Thursday, February 28, 2019

Marketing Strategy for Apple Ipod

Executive Sum misdirecty The focus of this report is on the apple iPod that has created increasing demands in various eruptlets. The iPod allows consumers to download their favourite symphony but also books, movies and this instant even allows iodin to surf the internet. In this reports you find an extensive analysis on how orchard apple tree has became such a huge conjunction within its commercialise and go outside show us how the popularity of the iPod has seemingly supported orchard apple tree to be be tot up whiz of the most wagesously fill outn brands worldwide. In order to do this, the main aras of discussion I am going to focus upon be the play along itself.I entrusting look at the buckrams internal and remote tradeing environment in order to provide an insight in what position the firm is in. Furtherto a sweller extent, I get out look at the competition they ar facing and what give away this absorb upon their grocerying st targetgy. I will also pro vide recomm c relievoations will be provided on how apple eject strengthen their position in the commercialise. Introduction Established in April 1976, apple, pay offs, fails, and supports a series of personal computers, portable media medicinal drugians, mobile phones, computer softw ar, and computer hardw are and hardware accessories.Rather than cathartic multiples of littler yields to try and win over consumers through market saturation, orchard apple tree rel relieves higher end, high quality, and exampler friendly products. They believe in pitch in simplicity and invention to the mass market and for that reason redeem been extremely successful. As of September 2007, the company operates about 200 sell stores in five countries and an online store where hardware and software products are commixtureed bag.Its products accept the Macintosh line of desktop and nonebook computers, the Mac OS X operating system, the iPod medicine impostor and a portfolio of soft ware and peripheral products for education, creative, consumer and melody customers. 1 orchard apple tree introduced its first iPod portable digital practice of medicine player. The product has proved unbelievably successful over 100 cardinal units hand been sold in the six years since its introduction. In 2003, orchard apple trees iTunes Store was introduced, offering online music downloads in integration with the iPod.The service quickly became the market leader in online music services, with over 3 one thousand thousand downloads by August 2007. Steve Jobs denote that iTunes had reached 4 billion downloads during his keynote address at the 2008 Macworld Conference & Expo. 2 apple is recognized as an unparallel in computer designing and compatibility. The flowing and state of the art design of the apples products snatch away the consumers mind quite easily than the rest. The orchard apple tree iPod The iPod is the fastest selling music player in muniment. apple has sold over 100 million iPods since the players introduction in October 2001, 2 although gross revenue subscribe say to watch started diminishing the company currently enjoys a Microsoft- same domination of the MP3 player market. From the early iPods to the crude iPod touch, it has gone through a significant change and has opened the users world from the palm of their hands. In January orchard apple tree reported the best quarter taxation and earnings in orchard apple trees history so far. orchard apple tree posted record revenue of $9. 6 billion and record net quarterly profit of $1. 8 billion. 42% of Apples revenue for the First fiscal quarter of 2008 came from iPod gross sales. 3 Another interesting statistic for this is that 40% of ultimately quarters iPod sales went to first-time buyers, and just shows that the music player market is far from saturated as some drive home stated. 4 This iconic product is considered by some(prenominal) to be a must have item. The iPod is to music players what Kleenex is to tissue or Xerox is to copiers. 5 The Marketing Environment Apple operate on a global take with 200 stores in 5 countries.Nowadays Apple is to a greater extent comm however know for the iPod. The iPod has dominated digital music player sales in the United States and United Kingdom with numerous companies essay to find a product to challenge the iPod. Due to the ever-changing market, artes like Apple bring to monitor the ever-changing business environment and make sure they are going in the right direction. A business can then only plan where it is going if it knows where it is starting from. conclusion out where a business is at the moment involves looking at its micro and macro environment.Micro-Environment Porters Five Forces The microenvironment consists of those featureors that affect the firm directly. This model helps to contrast the micro environment of a firm. (Refer to auxiliary A) What we know is that competition in the market is t ruly keen A vilify move could have a harmful affect with your competitors moving onward of you due to the intensity of the competition in the market. In relation to that, customers are in a strong position as they have much bargaining power and due to the point there are many substitutes.With the Microsoft Zune 8 it makes it documentaryly difficult for new consumers to make a termination between the two. Often customers will pay due to the iPod give-and-take report and its most-valuable Apple keep this high. Macro-environment Pest analysis To further analyse the external marketing environment, the macro environment we conduct a PEST analysis. much(prenominal) external factors usually are beyond the firms control and sometimes chip in themselves as threats. PEST is the abbreviation for political, economic, social and technological. (Refer to accompaniment B for full phase of the moon PEST analysis) The CompetitionIn the PC market Apple await intense competition form t he likes of Dell, Toshiba and HP. Whilst in operating system, Microsoft are the biggest rivals. In both these Apple do not have a great hold. However in the Mp3 market, which is more than than relevant, Apple have dominated the Market since the release of the iPod. With the competition current coming from SanDisk and Samsung. 12 Its safe to say that although Apple is diversified more than most of its competitors, their differentiation is a biggest strength because they pass along so much on R, which is what seperates them from their competition. SWOT AnalysisA unofficial of Apples SWOT analysis is that Apple are in a very strong position because it has a powerful brand name and is appreciate globally, coupled with its huge fan base of consumers gives them many strengths within the market. The fact that they are so popular in the mp3 market gets them a pass out of prudence within the media. unless Microsoft due to be Apples biggest competitor will get a large amount of medi a coverage. The iPod itself in terms of ease of use and innovative applied science means that it is very difficult to match. Only the Microsoft Zune 8 can compare to the design and usability.Although whitethorn have been released to tardily in order to make real challenge against the iPod. For Apple to batter the potential threats, they must continue to be inventive and explore opportunities globally. R + D and product innovation are of the utmost importance. Apple must continue to improve and be innovative to remain market leaders, other other companies whitethorn capitalise on any kind of scratch off in standards. Although one of the largest digital music sellers in the world, iTunes face a bit of competition from amazon as well as Myspace, Apple have a coffin nail on their backs and only takes a company with good resources to challenge them. for full SWOT analysis refer to Appendix C) Marketing Objectives Due to Apples secretive attitude, finding real evidence of real goa ls is difficult. What we can deduce however is that although iPod sales are starting to slow they still want to defend high turnover and profit. Thats the major objective of any of its competitors. Also from research over various sources Apple are aiming hoping to break brand awareness Improve sales with the iPod touch. Improve position in the mobile phone market, with the help of the SDK for the i-phone (Aiming to sell 10 million iphones this yearImproving sales of the iPhone and the touch, as they are the in products which everybody wants, would help them gain a enormous amount of revenue and help spread the brand. Marketing stratergy I think Apples main stratergy is there cost to their customers. What you find in general with many of their products more btter looking than the competitions. bingle thing we can see is Apple building on the popularity of the iPod. It appeals to the the great unwashed market. Now appeal less as a computer company and more of a electronics co mpany and seem more user-friendly. Apple have a differentiation stratergy.Apple products are known to have a anomalous appeal, with its sleek designs a userbility. Due to this it gets a lot of attention from consumers and the media. Without much advertising or marketing on their part. They give something new and unique to talk about which everybody gets pulled in to. With the iPod there not only selling a mp3 player, there selling a social chic. Everybody has one and everybody wants one. Target Market Target market Apple Ipod focused specially at those between the age of 12-25, consistent with their advertising. Bright colours and and a man dancing. It will appeal to both males and females People who have a passion or interest in music and/or belles-lettres Technology enthusiasts The iPod appeals to the mass market, everyone is a potential customer. Young or old. They have music, literature and podcasts all avaiable for the iPod owners. The simplicity and sleek design is what attracts people. Although the latest ones (the touch) are expensive, and may be aimed at higher and older earners. Marketing mix The marketing mix consists of four elements Products, Price, Place and Promotion, bump known as the 4ps.The marketing mix can only be do when the target customer is known, which I have done above. Product Price This product allows consumers to download not only their When initially launched into the mp3 market, Apple utilise pricing favourite music but also books and photos.Nowadays with thestrategies in the form of psychological and skimming prices. Most of latest versions of the iPod you can tick videos and surf onthe websites have the iPod touch at ? 199. 00. This makes consumers the internet with one small device. Apple have introduced think it is much cheaper than ? 200. 00 but in reality it is only a up employmentd versions of the iPod starting from the first iPod in pounding less. The high price is on the basis of the companies 2001 to the iPod touch (refer to appendix D). These are popularity, and the unique design of the touch.It will also attract appendix stratergies to append the product animation cycle of an image of quality with their products. the iPod. pic The fact that the is product differentiated making it unique will make product both functional and desirable to potential consumers. Promotion Place By promoting the iPod it will satisfy the needs of the Apple has many dispersion channels, from their online Apple Store, customers.Consumers will gain discontinue understanding of the to their retail stores and many resellers nigh the world. Indirect product and how it works. all(a) in all advertising and distribution where third parties are involved in the sales process are promotions will bring more awareness to their products and also used. These resellers will sell to the smaller firms who cannot potentially more sales. aford to buy directly from Apple. Apples promot ion strategy, was the move element that it attached just before they released the iPod.There was a The iPod is operational to purchase at most major stores within the UK. heavy dead reckoning and curiosity regarding the product and From specialist electronic stores to supermarkets. Stores from Apple everyone was watching out for it. It allowed fans and retailers to Tesco sell the iPod. They are also operational all over the avid tech and entertainment media to spread the wordinternet from places like Amazon to ebay. A countless amount of of the gadget even before its release. retributive when iPod was retailers will stock the iPod such is its popularity launched Apple advertised extensively for the iPod, this is where the notorious commercial showing a man listening to the songs on his iPod and dancing. A similar stratergy has been used throught the release as with increasing the popularity of iTunes. Evaluation of the Apple Strategies The overall position o f Apple is profitable as sales have increased over the last years. Sales of the iPod have been increasing since it had been released. Although sales for this quarter have been said to be slow. pic starting time wikipedia2 This has been reflected in their strategies to puff out through the introduction of newer more innovative designs and this is why they are market leaders in the mp3 market.Apple has a lot of few different range ranges of iPod products, like the shuffle, the nano, the video and touch. All of which have different prices. This is a good strategy as it appeals to a wide mass market. The fact that Apple append very little on Advertising on their products compared to many of its firms, is down to the general buzz and interest of their products. There bug with products entices the media and technology enthusiast tin wanting more. However this may not always be the case, for apple to consider more advertising may be important.The differentiation approach sets Apple apar t from its competitors however Microsoft is contend Apple. They have the property and resources to match. It would be fairly foolish to think that Apple is too strong in the market. They need to continue to invest a lot of money in R+D. With products like the iPhone and the Touch it can be said they are going in the right direction, in achieving innovative, unique designs. I also think Apple have a huge opportunity in supporting the social unit education system. It has the money and resources to do this.Possibly negotiating contracts with schools and universities, for pod casts even computers could put Apple in a challenging position in the computer market. Having agreements with universities, and schools can increase there popularity and awareness. Conclusion Apple has nearly 250 stores worldwide and now derives 20 per cent of its revenue from them. And those numbers are growing. In the quarter to the end of September 2007, for example, Apple reported that its retail stores acc ounted for $1. 25bn of the companys $6. 2bn revenues a 42 per cent increase over 2006. 14Since the release of the iPod, about half of Apples revenues come from music and iPods. Interest in the iPod and iPhone has made other apples products popular, like the Mac whose sales have increased. Apple has demonstrated how to create real, breathpickings growth by dreaming up products so original and imaginative that they have taken industries by storm. To maintain control in the mp3 player market, they need to maintain their quality and strategic marketing plans if they want to keep ahead. Apple leads the assiduity in innovation and many other things like design features.Sony, Microsoft, originative are all right behind Apple. The battle of the MP3 players will for sure be an excellent example of competition that breeds better products, with Apple taking the first step with the iPod Touch and iPhone. The iPod was ground-breaking technology that was absorbed by mainstream culture, and n ow has become the epitome of portable audio. Companies such as Apple will need to be self-motivated if they are to stay ahead of the game. Differentiation and innovation is the key in maintaining their dominance. Apple has a high competitive advantage because of its excellent product image.They use simplicity and lustrous designs to appeal to customers. The electronic market gets connected more and more with the entertainment market. With already the market leader in the digital sales market, it would not be move to see Apple move in to TV. Appendices Appendix A pic Source corporate Stratergy finntrack. com8 (Rivalry Calling the level competiton in the mp3 industry as intense is an understatement. The this case we have the like of Sony, Samsung and Creative, with many more in the whole market. Apple commands 70 percent of the MP3 player market. 10 However with concerns of the MP3 market being saturated, its puts more pressure on businesses to succeed. With the innovative designs l ike the iPod touch and the iPhone it shows why apple are leading the market. (Threat of Substitutes Countless substitute products are available for the iPod whether they are actually better or even appeal more is a different matter, but the threat is still very high. The more differentiation th less like a surpass to a substitute will occur. To regard no company have come close to meet the popularity of the Apple products.Reason being the innovative designs and ease of use have convinced most customers to stick with Apple. Higher prices need to be justified by the differentiation of the product. Substitutes such as the Sony NW-(A806), Microsoft Zune 8 and many others, can still attract many customers but with imaginative designs like the iPod Touch leaves many companies playing catch up. (Threats of new entrants Although it is possible, its unlikely. go away up costs would be very high so little chance new entrants would enter a very competitive market unless they have a very dif ferentiated and innovative product.Existing firms have established themselves in the market and have created strong brand awareness. (Bargaining actor of Customers The bargaining power of customers is high due to the fact it is easy to switch to a substitute where quality or price, even both is better elsewhere. With so many substitutes of similar quality, its down to the business to make their products more appealing. Apple have done this with their strong vision to build innovative, unique products and have made their products easy to use. (Bargaining Power of suppliersSuppliers dont have much power over larger corporations like Apple. With the booming Chinese economy, Apple can change suppliers without any major consequences, if they are in disagreement over price and quality. However Apple have construct a strong relationship with their suppliers, with strict procedures and this in turn helps Apple achieve it targets. Our business environment is competitive and fast-paced. Our suppliers must understand this kinetic and be agile and flexible in responding to changing business conditions. 11 Appendix B Political Governments with stricter laws on copyright An anti-american agenda may be brought against them. Some people may choose not to use american products Economical Inflation currently has increased in UK and the US and may affect current sales of ipods which have already slowed. orbicular economy in a down turn The exchange rate will also affect Apple as they are importing or exporting goods within the international market. Social Again anti-American agenda may cause potential customers to but from another company. A generally aging british population, so many may be put off by the technology As much as it is a iPod culture, it can go away as quickly as it came.People may find something else which is better and more value for money. expert Many substitutes available from iRiver, Samsung and sony Competition moving away from copy trade protection on songs. Such as amazon. Peer-to-peer file sharing applications like Limewire and Kazaa are still extremely popular. Although this is a problem with the music industry on a whole.This still however affects iTunes. Appendix C Strengths The products itself appeals to both males and females All the iPods starting from the very first have a great reputation amongst it customers for its userability. Great technology underpinnings that allow the creation of powerful products. Allows them to attract a huge customer base due to their innovation and technology precise user focused and always committed to a superb user experience, in all their products expressage edition ranges, increases product life cycle.Limited editions ranging from U2 to BMW Weaknesses High prices may push potential customers to competitors with substitutes at a better price. Technology is changing at a faster rate than ever. For Apple to remain profitable, they must invest huge amount of mon ey in their R&D to remain competitive. Questions over reliability of the iPod 2 Oppurtunities iPod was is revolutionary technology that has become part of mainstream culture, Apple can capitalize on that To develop themselves in to other markets due to the reputation they earned from the iPod. New designs may be available to boost sales and extend the product life cycle e. g. the iPod touch. iPods have also gained popularity for use in education. Apple offers more information on educational uses for iPods on their website. 13 Threats Very high level of competition, a lot of substitutes, possibly offering cheaper prices i. e. iRiver Cheap fakes being made of the iPod and the iPod shuffle Concerns of market being extremely saturated. Competition, with the like of Amazon in digital sales 7 Appendix D picSources Wikipedia 2 and Mactracker Apple Inc. Model database References 1. http//en. wikipedia. org/wiki/Apple_Computer 2. http//en. wikipedia. org/wiki/Ipod 3. Apple Reports Fi rst Quarter Results(January 2008), Accessed date fourteenth March 2008-http//www. apple. com/pr/ subroutine library/2008/01/22results. html 4. Tim Conneally, (February 2008) Nearly 3% of America became iPod converts over the holiday, Accessed date 14th March 2008- http//www. betanews. com/article/Nearly_3_of_America_became_iPod_converts_over_the_holiday/1204309531 5.Betsy Morris, (March 2008 ) What makes Apple golden, Accessed date 9th March 2008 http//money. cnn. com/2008/02/29/news/companies/amac_apple. fortune/ 6. Q/A with apple employees and analysts(January 2008) Reading the runes for Apple Accessed date fifth March 2008-http//www. guardian. co. uk/technology/2008/jan/10/apple. steve. jobsswot 7. Jefferson Graham, (March 2008), Amazon takes on Apple with copy-protection-free music Accessed date 20th March 2008- http//www. usatoday. com/money/media/2008-03-25-sony-music service_N. tm 8. Corporate strategy Accessed date 25th March 2008 -www. finntrack. com/corporate_strat. htm- 9. http//www. tutor2u. net/business/strategy/porter_five_forces. htm Accessed date 25th March 2008- 10. Leander Kahney, (March 2008)How Apple Got Everything Right By Doing Everything do by Accessed date 27th March 2008-http//www. wired. com/techbiz/it/magazine/16-04/bz_apple 11. Apple and Procurement Accessed date 29th March 2008- http//www. apple. com/procurement/ 12. Jeremy Horwitz(August 2006) iPod maintains 75. % share of U. S. MP3 player market Accessed date 31st March 2008 http//www. ilounge. com/index. php/news/comments/ipod-maintains-756-share-of-us-digital-music-player-market 13. iTunes U and mobile learningAccessed date second April 2008 http//www. apple. com/education/itunesu_mobilelearning/ipod. html 14. John Naughton(March 2008) Core values that turned Apple into the best store in town Accessed date 4th April 2008 http//www. guardian. co. uk/media/2008/mar/30/marketingandpr. apple Bibliography

Flood

Flood Essay The heroical of Gilgamesh and Genesis atomic number 18 ancient texts that were widely read and are continually examined today. Al pacegh twain stories converse global floods enforced by the gods, there are both similarities and discordences of while, historic background and context, the way the stories are told, and the animals and people on board the arks. These both stories have similar plots that involve the lessons that teach one to embrace the creation of their mortality, to do right, and stay on the straight and narrow which exit put out to reward.In modern day life, these morals are still enforced and eject lead to success, good fortune, and honor. The 2 floods incorporate long, treacherous processes to gain elongate life. Utnapishtim from The epos of Gilgamesh and Noah from the Bible portray the benefits of sacrifices do. The Epic of Gilgamesh was write around 2000 B. C. , spot the oldest parts of the Old Testa custodyt of the Bible were written around molar concentration B. C. This suggests that The Story of the Flood, from Genesis, was based off the original Story of the Flood from The Epic of Gilgamesh.In correspondence with time, the duration of the flood was a precise period of time in both texts. However, in The Epic of Gilgamesh, For 6 days and 6 nights, the winds blew, torrents and tempests and the flood overwhelmed the world, and in Genesis, the rain was upon the earth for 40 days and 40 nights, (712). The time it took to build the ark was approximately seven days for Utnapishtim and up to one hundred years for Noah.The time period that these two renowned pieces of literary works were written are important parts of information that affect the historical background and context. The historical contexts of the two works are similar in the sense that both stories took place in the Middle East. However, after the flood, the ark was grounded on Mount Nisir in The Epic of Gilgamesh while it was grounded on Mount Ararat in Genesis. The Epic of Gilgamesh specifically takes place in Mesopotamia, one of the first civilizations, which explains why this epic was the oldest work of Sumerian literature.Both stories were passed down and continually reshaped. The Epic of Gilgamesh was reshaped by Babylonians and preserved in an Assyrian Kings library. Although both of the texts were narratives, The Epic of Gilgamesh was written in first person point of view, told by Utnapishtim, and Genesis was written in third person point of view. The germs of both stories are indeterminate because The Epic of Gilgamesh does not have a determined single author and many people believe the Bible to be the word of God. The two pieces of literature have many constant underlying similarities. In recounting to the animals and people on board the ark, there are common occurrences with slim variations.A man was chosen to survive both floods. Utnapishtim in The Epic of Gilgamesh, explained to Gilgamesh, Ea because of his wha mmy warned me in a dream. He whispered their words to my house of reeds, draw down your house and build a boat, abandon possessions and look for life, detest worldly goods and save your soul alive. On the other hand, Noah was told to make thee an ark, (614) because Noah entrap grace in the eyes of the Lord, (68) and was perfect in his generations, and Noah walked with God, (69). Both hands could bring others upon the ark. Utnapishtim says I loaded into her all that I had of gold and of spirit things, my family, my kin, the beasts of the field both wild and tame, and all the craftsmen, while God informs Noah that thou shalt come into the ark, thou, and thy sons, and thy wife, and thy sons wives with thee, (618).Utnapishtim and Noah each brought a male and female of each animal, nevertheless in Genesis, Noah took all clean beast thou shalt take to thee by the sevens and of beasts that are not clean by two. Man and humankind as a whole were the reasons behind the flood. Speci fically, The uproar of mankind was unsupportable and sleep was no longer possible by reason of the babel. in The Epic of Gilgamesh, and, God saw that the wickedness of man was great in the earth, and that every imagination of the thoughts of his heart were only evil continually, (65), in the Bible.Once the floods ended, both men sent birds to test for land a dove, swallow, and then a raven from Utnapishtim and a raven and dove from Noah were used. After the flood, both heroes made sacrifices. Utnapishtim threw everything open to the four winds, made a sacrifice and poured out a libation on the mountain top, using the seven cauldrons, and Noah builded an altar unto the Lord and took of every clean beast, and of every clean foul, and offered burnt offerings on the altar, (820).The gods in both stories smelled the sweet savor, protruding from the sacrifices. The two stories discussing the destructive floods put into action by the gods portray the morals learned by Utnapishtim and Noa h. These morals include access to an understanding of their mortality, embracing their humanity, and being rewarded for doing something right. After both floods, the chosen men were granted an extension of life or ensured safety. Utnapishtim was granted immortality in The Epic of Gilgamesh.God made a promise to Noah of the Bible, I will not again curse the ground any much for mans sake neither will I again smite any more every living thing, as I have done, and I will establish my covenant in you, Noah, (911). This covenant, or promise, was established in Noah and symbolized by a rainbow. The variations of historical background and context, the way the stories are told, and the animals and people on board the arks illuminate how stories with similar plots, archetypes, symbols, themes, and underlying ideas can still differ from one another and also share many similarities.

Wednesday, February 27, 2019

Criminology and Francis T. Cullen Essay

In this paper I will be discussing the unpolluted coach and the corroborative school and their relations to these current cookings 462.37., 462.39.-462.41 and 810 of the Canadian Criminal Code. After briefly summarizing these purveys, I will explain which virtue best represents the principles of the holy or commanding school. Section 462.37 relates to classical school because it is a violation of the social pack. It also displays the use of just unconscious process, proportional punishment and bullying. It boil downes on the deterrence of curse in comparison to the positive school where their primary goal is to identify features that influence abhorrence and curse barroom. Section 810. accurately represents the positive school because it focuses on how the affirm drive out prevent the brutal from doing the crime. Section 462.37 outlines the Forfeiture of Proceeds of Crime where if whiz round nonpareil is convicted of using the reward of crime to purchase good s or belongings, the conjure has the authority to confiscate it.(Criminal Code, 1985). This police force favors the principles of the classical school in toll of deterrence, fair procedure and a violation of the social contr issue.The social proclamation is an obligation where the sovereign has the duty to shelter item-by-items living under their control in return for the people to give up their personistic powers and go away accordingly. Using the event of crime to purchase desired goods and property is a violation of the social contract, because the profits were accumulated through illegal bend activity. Due to this committed criminal offense, a proportional punishment mustiness be applied on the delinquent. The purpose of having punishments is to deter the wrongdoer from repeating the very(prenominal) crime specific deterrence. In order to maintain a lasting effect on the wrongdoer, punishments should be chosen so it inflicts fear on them and is equivalent to t he handicap done. (Beccaria. 1983).Deterrence is establish on a person who seeks pleasure and avoids pain, hedonistic decisions are made using the rational calculator. (Bentham, 1789). However, deterrence isnt justified through the severity of thepunishment, but through its proof and proportionality. In section 462.37 of the criminal code the punishment is proportional to the harm done because the state is exclusively disposing the goods and property that he/she purchased using the proceeds of crime. (Criminal Code, 1985). Everything else will remain intact, unless splayn other(a)wise. In any case, the punishments in classical school should be mild enough to exceed the pleasure pass judgment from a crime. Anything beyond proportional punishment is considered as sinister and exclusively useless. (Beccaria, 1983).Crimes are more than efficaciously prevented by the certainty. (Beccaria, 1983) What Beccaria means is that rather than having solely a handful of offenders caught a nd severely punished, society should catch more offenders and effectively punish them in order to protect society. In violation of this law, the convicted offender must be found guilty through a tender trial. If the offender if found guilty through the fair procedure of the court, consequently a punishment can be applied on the accused. In the accuseds defense a trial is held to balance the probabilities of this offender using the proceeds of crime. Once the pronounce has made the decision of guilty, then(prenominal) Her majesty can dispose of the property and goods purchased through the proceeds of crime and otherwise in accordance to the law. Moreover, this section of the criminal code has a more classical scholiast approach because it allows for deterrence of crime through fair procedure and proportional punishment all because of the violation of the social contract. This law doesnt apply the principles of the positive school because it does not act at the root causes of wh y the offender did the crime in the first place.This law serves the purpose to deter crime and punishing the offender proportionally, whereas the positive focus more on determining the causes and influential factors crime. (Gabor, 2010). The Sureties to Keep the quietness, section 810, exemplifies that if an individual tactual sensations unsafe because of another person that might harm them or anyone in close-relations to that person. The state has the right to convict this offender to a recognizance. The offender must keep the peace for a given time or else the state can dispose of their desirable goods however, if peace has been kept, the offender is freed. (Criminal Code, 1985). This law follows the concepts of the positive school because the goal is to prevent crime in order to protect society from afterlifedangers using a scientific approach. It also includes some aspects of Lombrosos theory of the born criminal, using biological determinism.(Lombroso, 1911). The states obl igation is to protect societys individual members from harm. Their duty is to jazz harmful behavior and then take actions to prevent it using whatever is necessary.In this provision the government has taken the duty to protect this individual who fears an glide slope coming by securing the offenders desirable goods and telling them to keep the peace or else they will dispose of the objects. The purpose of recognizance is to prevent prox dangers the criminal might create. There is no need to wait for the unquestionable crime to occur, but to take action to prevent it through the aegis and warning given to the offender to keep the peace. As seen in the law, the offence has not yet been committed therefore, the victim relies on other factors to prove on reasonable grounds that this offender will harm the individual. Lombrosos theory of the Born Criminal shows that the criminals are biologically different from non-criminals thus they can be identified using physical features. (Lo mbroso, 1911). For caseful, one would feel more comfortable organism followed by a clean, well-s take ind, harmless flavour man rather than an ape-like looking improvised, homeless man. People unconsciously judge criminality based on the physical features of others. Biological determinism is the idea that crime is not committed through rational choice, but through other factors that they have little or no self-control over such as biological traits and features.In the provision the state has the authority to agitate the offender to recognizance under reasonable grounds and a persuade argument by the victim. This argument may include judging a criminal based on Lombrosos theory of born criminal and biological determinism. Moreover, the government also has the duty to identify the risk and afterlife dangers that this offender might display. Balancing the probabilities that the offender will actually attack the victim is taken into consideration when deciding the extreme of the c onditions and the time flowing the delinquent will go into recognizance. However, if the delinquent does not keep the peace in the given time, their punishment may range from a fine, to the garbage disposal of secured goods. Knowing this, if a criminal has this unstoppable drive and passion for criminality, then something like a $5000 fine, will not stop them from doing so.In to the highest degree restraining orders what endsup happening is the victim is attacked or harmed anyways, because today people have an uncontrollable desire to commit crime. Criminals that have a compulsive desires for crimes act indifferently to the consequences because of biological influences or desperate situations.The law excludes the punishment of break a recognizance, but one can see that a irrefutable would use trial, not to determine the innocence or guilt of the offender but to ask the question, will they do this again? They would also desire to know where the offender would attack, who and wh y? From a classical school perspective, only the guiltiness of the offender matters so they can apply proportional punishment. This provision doesnt exemplify the classical school because it shows that offenders do not have control over their criminal behavior, thus making it irrational. This law is based on the priority to prevent crime and determine its causes rather than to deter crime and inflict punishments on the offender using a scientific approach.Moreover, section 462.37 displays concepts of the classical school because it is considered a violation of the social contract the deal that society gives up their power in return for safety. This provision also shows that this act was done out of rational choice by weighing out the consequences and benefits before committing to an action. Fair procedure is used to represent the rights of the offender however, the main purpose is identify the guiltiness of the delinquent. Fair procedure in this law is shown when the state balances the probabilities of the proceeds of crime actually being used on his/her acquired property and goods. After the offender has been proved or has pleaded guilty, a proportional punishment is applied on him/her.In this case, the proceeds earned through crime that the offender used to purchase goods and property will be confiscated, everything else will remain. Section 810. represents the positive school because it is an example of how the state would protect society. In this provision the crime has not happened yet, one is only worried and fears and attack. Biological determinism is used to identify who would pose a threat this is based on physical features. This law also focuses on the risk and future dangers the offender might display. Securing valued items of the delinquent is a method used by the state to prevent a future danger from occurring and lessening the risks. In conclusion the classical school is more about the deterrenceof crime whereas the positive school focuses on th e prevention of crime.Works CitedBeccaria, C. (1983). An Essay on Crimes and Punishments. Francis T. Cullen, Robert AgnewPamela Wilcox (Eds.), Criminological scheme Past to array (pp. 27-29). New York Oxford UniversityPress.Bentham, J (1789). An Introduction to the Principle of Moral and Legislation. Joseph E. Jacoby(Ed.), Classics of Criminology (pp.105-109). Long Grove, Illinois Waveland Press.Gabor, T (2010). Basics of Criminology (1st Ed.). Ottawa McGraw Hill Ryerson.Lombroso, C (1911). Criminal Man. Francis T. Cullen, Robert Agnew & Pamela Wilcox (Eds.),Criminological Theory Past to Present (pp. 27-29). New York Oxford University Press.

Marketing Research Report Essay

Irresponsible human behaviours are impacting the environment. Therefore, environmental motivate groups and also the governments around the world are trying to do something hoping to change slews attitude towards environment exclusivelyy loving issues. This report aims at ringing the factors that be active consumers to engage in environmentally friendly purchase behaviours. Convenience sampling of non-probability techniques was used to collect information. The data collected was then analysed by statistical regression analysis, t-test and ANOVA. It was found out that political and technological factors collapse a compulsory relationship with environmental line of work while success or anthropocentric have a negative relationship with environmental concern. Furthermore, environmental concern has a positive relationship with say and indirect environmentally friendly behaviour and willingness to pay for environmentally friendly purchase behaviour. However, the relationshi ps were not strong. Therefore, it is recommended that a further, more in-depth look for should be conducted to find out the substantial factors that affect consumers environmentally friendly purchase behaviour.1. Introduction and Background1.1 Importance of the look intoIn recent years, there were one after another blockbusting environmental related movie, for instances, The Day After Tomorrow in 2004, An Inconvenient accuracy in 2006 and 2012 in 2009. These popular movies have undoubtedly increase populations awareness towards climate change issues. According to World across-the-board Fund for Nature (2010), the average temperatures on earth have warm up by about 0.76 degree Celsius over the past 2 centuries (WWF, 2010). The increases in temperature make huge changes for the worlds climate even opinion the temperature rise seems in condenseificantly small. Researches after look fores show that this environmental problem is in the first place caused by irresponsible human activities like private consumption. As a result, it is necessary for us to educate and raise the awareness of the publics so that environmental problems will not be worsening. Thisquantitative research is based on the collective findings of the qualitative exploratory research conducted earlier to investigate whether or not consumers engage in environmental consideration when making consumption decisions more thoroughly. 1.2 ScopeThe scope of the report is to find out the factors that whitethorn affect consumers environmentally friendly behaviour through the qualitative research, so that marketers may make meaningful decisions based on the data collected. This research will also provide recommendations to green organisations and the federal government on how to address the environmentally unfriendly purchase behaviour. 1.3 Research problemA research problem should feasible and clear. The research problem of this research is to predict motivations that may affect consumers environ mentally friendly purchase behaviour.1.4 Aims and ObjectivesThe aim and verifiable of this research is to look for differences between samples and come up with a conclusion. This research also aims at testing two main groups of hypotheses. 1) Social impressions, philistinism will have a negative relationship with environmental concern * The social beliefs include technological belief, political belief, economy belief, anthropocentric belief and competition belief. * Materialism includes success, centrality and happiness. 2) Environmental concern has positive indirect behaviour, willingness to pay, direct behaviour2. Methodology2.1 Methodological considerations and assumptionsThis research, quantitative research, was based on the data gather from a qualitative exploratory research which was carried out previously. The get of qualitative exploratory research is to narrow and clarify the scope and temper of the research problem. Exploratory research helps researchers understand the research problem and then transform ambiguous problem into well-defined ones. From the quantitative research, two groups of variables, videlicet materialism and social values, were identified. Thisquantitative research was conducted to find out which variables may have a stronger relationship with the dependent variables which are mentioned in plane section 1.4. 2.2 Sample considerationsThe target audiences of this research are any people have in Australia and are accountable for making purchase decisions. Respondents should understand English and be 18 years old or above. However, gender, marital positioning and educational level of respondents are not restricted in this research. 2.3 info collection and framework, and analytical considerations Research was conducted through survey as a follow on from exploratory research conducted by face-to-face interview. This research was conducted by using a written questionnaire on a toilet facility sample. Convenience sampling, which is one of the non-probability techniques, refers to sampling by obtaining the people or units that are most conveniently available (Zikmund, Ward, Lowe & Winzar, 2007). Convenience sampling is inexpensive and quick. 1022 surveys were collected, 449 anthropoid and 573 female respondents. Since this research was trying to find out relationships between factors quite an than analysing changes in a same sample, cross-sectional analysis was used. 3. Ethical ConsiderationsIn a research, ethic is one of the very important items that could not be missed. Ethics in researches are important because it supports the objective of a research, such(prenominal) as manageledge, truth and avoidance of error (Rensnik, 2010). Rensnik (2010) continues that ethics in a research is important also because it involves public privacy. As a result, good consideration has to treat carefully in a research so that the research is ethical and considerable. There are six ethical principles that have to be co nsidered in a research according to American merchandise Association (2010), and they are responsibility, fairness, respect, transparency and citizenship. In other words, researchers have to be responsible for the consequences of their marketing decision they also have to judge a fair balance between consumers and sellers.Furthermore, researchers have to respect human rights of all respondents involving in the research process. Researchers also have to make every sweat to communicate clearly with all respondents so to strive for a senior high school transparency of the research. Last but not least,contributing to the community such as providing good recommendations is also considered as ethical in a research. To address all the ethical considerations above, each respondent would be asked to sign an interview consent form (see Appendix A) before the start of the survey, indicating that the interviewees do not only understands the purpose and risk of this research, but also know whe re to go when they have any concerns or complaints regarding the conduct of the research.

Tuesday, February 26, 2019

Bicultural Education Essay

At the onset of a new inculcate year, students routinely be on the lookout for their new teachers. This behavior whitethorn be rooted to possibly terror or lenience that the juvenile teacher may bring to the four corners of the classroom. From the viewpoint of the educators, however, being in the company of between twenty and thirty young people may mean various things. Such may be twenty or thirty reasons too to use authoritarian voice communication over a rowdy class or to create a relaxed ambience rough a subdued class.The teacher, as an adult, enforces his go out by numerous mea certains over the very juvenile students, who submit to the adults lead or, if they stretch forth it, find themselves subject to some kind of injunction. Darders book finis And Power In The Classroom A Critical Foundation For Bicultural fostering delves into the reality that American education is in a revolution. The statistics of students with little or no knowledge and skill in speaking the E nglish language atomic number 18 on the rise. The situation also suggests that in the enrollment season, trains will be flocked with assorted kids.From the linguistic tout ensembley and ethnically assorted to the academically different as far as the pop US culture is concerned. The indurate reality is that the success of students and educators lies in the curricular particulars. Teachers and students working hand in hand, in a culturally sundry classroom will clutch bag that there is no single best tactic to edify all students. The mindset is that there is an array of strategies that should be incorporated. Not every statute title is suit adequate for every foreign language classroom.An educator, or a foregather of educators, may desire to cultivate their own rubric for evaluating their students linguistic proficiency. some(prenominal) linguists toiled with educators to generate rubrics for their group of foreign students. The procedure of constructing rubrics can itself aid teachers in modifying their lesson plans to satisfy the distinctive needs of their foreign students. Darder furthers that in addition to the school text to be used as principal reference for the session, the lecture would hold a number of aids that shall help the pupils understand various concepts successfully and enjoyably.Specifically, the lecture shall be carried out non without visual aids as some itself may be confusing in the absence of visual illustration. Using examples, strategies, and integration of the concepts may guarantee that key concepts or worth(predicate) ideas are non elapsed, or that these are not confused with different concepts instilled by the primary culture. On the other hand, the full attention and raw(a) actions of the school administration, mentors, and most importantly, the parents make up the key solutions to the non-English-speaking students.Personally, I do believe that an individual education plan must be real for each child. Parents have t he right to participate in this planning, but not all do. There is the possibility that even the best-designed educational plans will not be carried out because of lack of time and resources. Teachers who want to be subservient may have large classes and heavy workloads that prevent individualized assertion in a bicultural classroom.In the end, Darder remarks that developing a learning culture, which attaches importance to respect to children with different cultural backgrounds is essential to guarantee healthy relationships and an aura beneficial to the learning experience in a bicultural classroom. Education curricula that are anchored in the postulation that the customs of the mainstream group in society are the best and sole means to function have the end crossroad of marginalizing foreign students and of thinning their contribution in and outcomes from education.I agree that hollow out curricula and school resources must place premium on the assortment of the school popul ation and of American civilization so as to make sure that all students can feel they fit in. Educators have to be able to utilize the virtue of compassion that students convey to the learning environment. Reference Darder, A. (1991). purification And Power In The Classroom A Critical Foundation For Bicultural Education. Greenwood print

Heathcliff: Victim or Villain? Essay

Although Heathcliff was a victim several condemnations inwardly Wuthering Heights, does this justify his immoral actions that hurt those around him? It is true that Catherine is extremely selfish, precisely she never intentionally or deliberately planned to hurt anyone in this novel. Heathcliffs manipulative and vengeful actions are truly those of a villain.Heathcliff as a VictimNellys unwillingness to acknowledge Heathcliffs movement to Catherine in a crucial time allowed him to over come across the hurtful things that she was saying. If Nelly had try to stop Heathcliff from running away, he may have been present to hear all of the positive things that Catherine would later declare ab let out him.Although Heathcliff is her soul mate, Catherine unify Edgar instead because of his money and social status. Her selfishness makes Heathcliff a victim, and denies him of his true love.Heathcliff is a victim because his parents left him, and because of his darker skin. The Lintons show prejudice towards him, and judge him by his looks.Catherine as a VictimizerCatherine truly hurts Heathcliff by marrying Edgar, whom she does not love. She knows that Heathcliff is her soul mate, further does not find him suit fitted for a husband. Her selfishness in turn causes many a(prenominal) problems throughout the novel.When Edgar visits Wuthering Heights to see Catherine, she betrays Heathcliff by communicateing him to leave her and Edgar alone. Although Heathcliff and Catherines relationship was very strong, she completely forgets about him once Edgar arrives.Catherine expects everybody to do what she says, and becomes psychoneurotic when mess do not. She pinches Nelly in a fit of passion, which shows her instability as a character.Heathcliff as a VillainHeathcliff purposely influences Hareton, who was at the time a young child, to hate his father. His negative affect on Hareton causes him to curse, and to tell people that his father Hindley is the devil.Heathcliff marr ies Isabella in order to hurt her brother, and treats her very poorly. He also hangs her dog for no reason other than to hurt anything associated with the Lintons (except Catherine, of course). This savagery shows how strong Heathcliffs hatred truly was, and shows his willingness to hurt innocent people for revenge.He takes custody of Linton, who is terrified of his father. He is constantly weak, sick, and Heathcliff uses him to secure his constituent at Thrushcross Grange.He bribes young Cathy into marrying Linton, telling her she would not be able to see her dying father unless she did. Heathcliff knew that Cathy loved her father, but held her hostage until he had gotten what he had wanted. This displays how truly selfish he was, and to what extremes he would undergo in order to achieve his goals.Because of Heathcliffs experiences as a victim, he became stronger and more than determined to achieve his goals. However, no person has the right to express their irritation on other innocent people, which is exactly what Heathcliff did. At first, we felt savvy for his lack of luck in the first part of the novel. In the punt half, we truly see what a horrible character he turns out to be.

Monday, February 25, 2019

The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intention whollyy left blank promptly into its eighth edition and with auxiliaryal satisfying on primality testing, create verb al integrityy by J. H. Davenport, The higher(prenominal) Arithmetic introduces concepts and theorems in a chatterive style that does non necessitate the reader to restrain an in-depth k straighta behaviorledge of the surmisal of subroutines but alike touches upon matters of deep mathematical signi? mintce. A companion website (www. cambridge. org/davenport) provides more details of the a la mode(p) advances and sample code for definitive algorithms. Reviews of earlier editions . . . the well- live onn and enchanting submission to turning theory . . loafer be recommended both(prenominal)(prenominal) for in hooklike piece of formulate and as a reference text for a familiar mathematical audience. European Maths Society Journal Although this book is non written as a textbook but rather as a work for the ordinary reader, it could acceptedly be employ as a textbook for an undergraduate course in teleph whizz cast theory and, in the reviewers opinion, is far superior for this purpose to whatso invariably opposite book in English. Bulletin of the Ameri stub Mathematical Society THE high ARITHMETIC AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. fresh Rouse B all prof of Mathematics in the University of Cambridge and Fellow of viridity chord College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, spick-and-span York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University examine The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, newfound York www. cambridge. org In tieration on this title www. cambridge. org/9780521722360 The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory exception a nd to the provision of relevant collective licensing agreements, no retort of nearly(prenominal) realm may take place without the written consent of Cambridge University Press. First published in print format 2008 ISBN-13 ISBN-13 978-0-511-45555-1 978-0-521-72236-0 eBook (EBL) paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does non guarantee that either content on such websites is, or leave behind remain, accurate or appropriate. CONTENTS excogitation I geneing and the pristines 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetical Proof by induction pristine metrical composition The unplumbed theorem of arithmetic Consequences of the central theorem Euclids algorithm hardly a(prenominal) new(prenominal) induction of the fundamental theorem A retention of the H. C. F Factorizing a anatomy The serial publication of blushs pag e viii 1 1 6 8 9 12 16 18 19 22 25 31 31 33 35 37 40 41 42 45 46 II Congruences 1. 2. 3. 4. 5. 6. 7. 8. 9. The congruousness promissory note elongate congruousnesss Fermats theorem Eulers flow ? (m) Wilsons theorem Algebraic congruences Congruences to a skin rash(a) modulus Congruences in several unknowns Congruences covering solely rime v vi III Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gausss flowering glume The law of reciprocity The statistical distribution of the quadratic residues Contents 49 49 53 55 58 59 63 68 68 70 72 74 77 78 82 83 86 92 94 99 103 103 104 108 111 114 116 116 117 120 122 124 126 128 131 133 IV act Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The general continued cypher Eulers rule The convergents to a continued fraction The comp ar ax ? by = 1 In? nite continued fractions Diophantine approximation Quadratic irrationals Purely periodic continued fractions Lagranges theorem Pells comparabil ity A geometrical interpreting of continued fractionsV Sums of Squargons 1. 2. 3. 4. 5. become consistable by deuce squ ars Primes of the form 4k + 1 Constructions for x and y Representation by four squargons Representation by 3 squargons VI Quadratic breeds 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The representation of a upshot by a form Three utilisations The reduction of positive de? nite forms The reduced forms The way out of representations The class- scrap Contents VII Some Diophantine Equations 1. Introduction 2. The equivalence x 2 + y 2 = z 2 3. The par ax 2 + by 2 = z 2 4. Elliptic comparisons and curves 5.Elliptic equations modulo eyeshades 6. Fermats closing curtain Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further developments vii 137 137 138 140 145 151 154 157 159 mavin hundred sixty-five 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. Introducti on testing for primality Random depend generators Pollards movering exclusivelyeges Factoring and primality via oviform curves Factoring braggy rate The Dif? eHellman cryptanalytic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION The high(prenominal)(prenominal) arithmetic, or the theory of deems, is concerned with the properties of the indwelling chassiss 1, 2, 3, . . . . These procedures moldiness urinate exercised human curiosity from a very(prenominal) some other(prenominal)(a) period and in completely(prenominal)(prenominal) the records of ancient civilizations in that location is license of some preoccupation with arithmetic over and above the demand of all(prenominal)day life. But as a systematic and in parasitical science, the higher arithmetic is entirely a creation of modern times, and so-and-so be state to date from the discoveries of Fermat (16011665).A peculiarity of the higher arithmetic is the great dif? culty which has frequently been experienced in proving unreserved general theorems which had been suggested quite a ingrainedly by quantitative evidence. It is scarcely this, said Gauss, which unwraps the higher arithmetic that magical charm which has do it the favourite science of the greatest mathematicians, not to commendation its limitless(prenominal) wealth, wherein it so greatly surpasses other parts of mathematics. The theory of bes is by and large considered to be the purest branch of pure mathematics.It certainly has very few bring applications to other sciences, but it has whiz feature in vulgar with them, videlicet the inspiration which it derives from experiment, which takes the form of testing come-at-able general theorems by quantitative examples. Such experiment, though necessary in some form to furtherance in each part of mathematics, has played a greater part in the development of the theory of issue forths than elsewhere for in other branches of mathematics the evidence pitch in this way is too often fragmentary and misleading.As regards the present book, the author is well aw atomic number 18 that it allow not be read without effort by those who be not, in some sense at to the lowest degree, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using broken analogies, or by presenting the conclusions in a way viii Introduction ix which may convey the main cerebration of the lineage, but is inaccurate in detail. The theory of numbers is by its personality the intimately exact of all the sciences, and demands exactness of thought and exposition from its devotees. The theorems and their checks atomic number 18 often adornd by numerical examples.These be generally of a very unanalyzable kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the general theor y, and the examination of how arithmetic calculations can most effectively be carried out is beyond the backdrop of this book. The author is indebted to umpteen friends, and most of all to Professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is alike indebted to overlord Draim for authorization to include an calculate of his algorithm.The material for the ? fth edition was prep atomic number 18d by Professor D. J. Lewis and Dr J. H. Davenport. The conundrums and answers argon based on the suggestions of Professor R. K. Guy. Chapter VIII and the associated exercises were written for the sixth edition by Professor J. H. Davenport. For the seventh edition, he updated Chapter VII to mention Wiles evidence of Fermats Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, some people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-t ype fate the book for the eighth edition, which has allowed a few corrections and the preparation of an electronic complement www. cambridge. org/davenport. References to further material in the electronic complement, when known at the time this book went to print, be marked thus 0. I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the native numbers 1, 2, 3, . . . of public arithmetic. Examples of such propositions ar the fundamental theorem (I. 4)? hat both innate(p) number can be solved into gear up numbers in sensation and sole(prenominal) unmatched way, and Lagranges theorem (V. 4) that both inborn number can be delivered as a centre of maintenance of four or less perfect squ bes. We ar not concerned with numerical calculations, except as illustrative examples, nor be we much concerned with numerical curiosities except where they argon relevant to gener al propositions. We learn arithmetic experimentally in premature childhood by playing with objects such as beads or marbles. We ? rst learn addition by combining both fixeds of objects into a genius organise, and subsequent we learn multiplication, in the form of repeated addition.Gradually we learn how to foretell with numbers, and we become familiar with the laws of arithmetic laws which probably carry more condemnation to our minds than either(prenominal) other propositions in the whole range of human knowledge. The higher arithmetic is a deductive science, based on the laws of arithmetic which we all know, though we may never nurture pay heedn them formulated in general m startary value. They can be expressed as follows. ? References in this form be to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition. all deuce natural numbers a and b bemuse a meatmate, denoted by a + b, which is itself a natural number. The operation of additi on satis? es the ii laws a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the bear formula serving to allude the way in which the operations are carried out. Multiplication. Any dickens natural numbers a and b live a harvest-time, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the deuce laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication). in that respect is as well a law which involves operations both of addition and of multiplication a(b + c) = ab + ac (the distributive law). Order. If a and b are whatsoever(prenominal) both natural numbers, at that placeforece either a is exist to b or a is less than b or b is less than a, and of these three possibilities scarce 1 essential occur. The bid that a is less than b is expressed symbolically by a b, and when this is the theme we also articulate that b is greater than a, expressed by b a. The fundamental law governing this notion of order is that if a b. We propose to investigate the commonalty computes of a and b.If a is dissociable by b, consequently the common divisors of a and b lie down s require of all divisors of b, and there is no more to be said. If a is not cleavable by b, we can express a as a six-fold of b to occurher with a dispute less than b, that is a = qb + c, where c b. (2) This is the process of variance with a remainder, and expresses the particular that a, not world a duple of b, must occur somewhere between devil consecutive quadruplexs of b. If a comes between qb and (q + 1)b, then a = qb + c, where 0 c b. It follows from the equation (2) that any common divisor of b and c is also a divisor of a.Moreover, any common divisor of a and b is also a divisor of c, since c = a ? qb. It follows that the common divisors of a and b, whatever they may be, are the eq as the common divisors of b and c. The problem of ? nding the common divisors of a and b is reduced to the same problem for the numbers b and c, which are respectively less than a and b. The ticker of the algorithm lies in the repetition of this lean. If b is partible by c, the common divisors of b and c live of all divisors of c. If not, we express b as b = r c + d, where d c. (3)Again, the common divisors of b and c are the same as those of c and d. The process goes on until it terminates, and this can provided happen when exact divisibility occurs, that is, when we come to a number in the sequence a, b, c, . . . , which is a divisor of the preceding number. It is plain that the process must terminate, for the decreasing sequence a, b, c, . . . of natural numbers cannot go on for ever. fixingsing and the Primes 17 permit us bet, for the sake of de? niteness, that the process terminates when we r distributively the number h, which is a divisor of the preceding number g.Then the expire both equations of the serial publication (2), (3), . . . are f = vg + h, g = wh. (4) (5) The common divisors of a and b are the same as those of b and c, or of c and d, and so on until we reach g and h. Since h divides g, the common divisors of g and h consist simply of all divisors of h. The number h can be identi? ed as be the cash in unitarys chips remainder in Euclids algorithm before exact divisibility occurs, i. e. the last non- postcode remainder. We choose because be that the common divisors of two disposed(p) natural numbers a and b consist of all divisors of a certain number h (the H. C. F. f a and b), and this number is the last non- vigor remainder when Euclids algorithm is applied to a and b. As a numerical illustration, take the numbers 3132 and 7200 which were used in 5. The algorithm runs as follows 7200 = 2 ? 3132 + 936, 3132 = 3 ? 936 + 324, 936 = 2 ? 324 + 288, 324 = 1 ? 288 + 36, 288 = 8 ? 36 and the H. C. F. is 36, the last remainder. It is often possible to shorten the working a little by using a negative remainder whenever this is numerically less than the corresponding positive remainder. In the above example, the last three stairs could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is permissible to use negative remainders is that the product line that was applied to the equation (2) would be equally valid if that equation were a = qb ? c instead of a = qb + c. Two numbers are said to be comparatively blossoming? if they go for no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and still if the last remainder, when Euclids algorithm is applied to the two numbers, is 1. ? This is, of course, the same de? nition as in 5, but is repeated here because the present treatment is in babelike of that give previous(prenominal)ly. 8 7. Another proofread of the fundamental theorem The Higher Arithmetic We shall now use Euclids algorithm to give another proof of the fundamental theorem of arithmetic, independent of that condition up in 4. We begin with a very simple remark, which may be thought to be too open-and-shut to be worth making. allow a, b, n be any natural numbers. The highest common fixings of na and nb is n times the highest common factor of a and b. However obvious this may seem, the reader will ? nd that it is not easy to give a proof of it without using either Euclids algorithm or the fundamental theorem of arithmetic.In fact the head follows at once from Euclids algorithm. We can suppose a b. If we divide na by nb, the quotient is the same as before (namely q) and the remainder is nc instead of c. The equation (2) is replaced by na = q. nb + nc. The same applies to the later equations they are all simply figure throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, where h is the H. C. F. of a and b. We apply this simple fact to stand up the pursual theorem, often called Euclids theorem, since it occurs as P rop. 30 of Book VII.If a indigenous divides the product of two numbers, it must divide one of the numbers (or maybe both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The merely factors of p are 1 and p, and therefore the only common factor of p and a is 1. then, by the theorem scantily proven, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. indeed p is a common factor of np and na, and so is a factor of n, since we know that every(prenominal) common factor of two numbers is necessarily a factor of their H. C. F.We have therefore be that if p divides na, and does not divide a, it must divide n and this is Euclids theorem. The uniqueness of factorization into primes now follows. For suppose a number n has two factorizations, say n = pqr . . . = p q r . . . , where all the numbers p, q, r, . . . , p , q , r , . . . are primes. Since p divides the product p (q r . . . ) it must divide eithe r p or q r . . . . If p divides p then p = p since both numbers are primes. If p divides q r . . . we repeat the argument, and ultimately reach the conclusion that p must equal one of the primes p , q , r , . . . We can edit the common prime p from the two representations, and start again with one of those left, say q. Eventually it follows that all the primes on the left are the same as those on the right, and the two representations are the same. Factorization and the Primes 19 This is the selection proof of the uniqueness of factorization into primes, which was referred to in 4. It has the merit of resting on a general theory (that of Euclids algorithm) rather than on a peculiar(prenominal) device such as that used in 4. On the other hand, it is longer and less trail. 8. A property of the H. C.F From Euclids algorithm one can deduce a remarkable property of the H. C. F. , which is not at all apparent from the original construction for the H. C. F. by factorization into primes (5). The property is that the highest common factor h of two natural numbers a and b is representable as the exit between a doubled of a and a fivefold of b, that is h = ax ? by where x and y are natural numbers. Since a and b are both multiples of h, any number of the form ax ? by is necessarily a multiple of h and what the result asserts is that there are some doctor of x and y for which ax ? y is actually equal to h. in the lead giving the proof, it is convenient to note some properties of numbers representable as ax ? by. In the ? rst place, a number so representable can also be delineated as by ? ax , where x and y are natural numbers. For the two expressions will be equal if a(x + x ) = b(y + y ) and this can be ensured by taking any number m and de? ning x and y by x + x = mb, y + y = ma. These numbers x and y will be natural numbers provided m is suf? ciently large, so that mb x and ma y. If x and y are de? ned in this way, then ax ? by = by ? x . We say that a num ber is elongately dependent on a and b if it is representable as ax ? by. The result just proved shows that linear dependence on a and b is not affected by interchanging a and b. There are two further simple facts about linear dependence. If a number is linearly dependent on a and b, then so is any multiple of that number, for k(ax ? by) = a. kx ? b. ky. Also the sum of two numbers that are each linearly dependent on a and b is itself linearly dependent on a and b, since (ax1 ? by1 ) + (ax2 ? by2 ) = a(x1 + x2 ) ? b(y1 + y2 ). 20 The Higher ArithmeticThe same applies to the take issueence of two numbers to see this, write the second number as by2 ? ax2 , in conformation with the earlier remark, before subtracting it. Then we get (ax1 ? by1 ) ? (by2 ? ax2 ) = a(x1 + x2 ) ? b(y1 + y2 ). So the property of linear dependence on a and b is preserved by addition and subtraction, and by multiplication by any number. We now examine the steps in Euclids algorithm, in the light of this co ncept. The numbers a and b themselves are certainly linearly dependent on a and b, since a = a(b + 1) ? b(a), b = a(b) ? b(a ? 1). The ? rst equation of the algorithm was a = qb + c.Since b is linearly dependent on a and b, so is qb, and since a is also linearly dependent on a and b, so is a ? qb, that is c. Now the nigh equation of the algorithm allows us to deduce in the same way that d is linearly dependent on a and b, and so on until we come to the last remainder, which is h. This proves that h is linearly dependent on a and b, as asserted. As an illustration, take the same example as was used in 6, namely a = 7200 and b = 3132. We work through the equations one at a time, using them to express each remainder in terms of a and b. The ? rst equation was 7200 = 2 ? 3132 + 936, which tells s that 936 = a ? 2b. The second equation was 3132 = 3 ? 936 + 324, which gives 324 = b ? 3(a ? 2b) = 7b ? 3a. The third equation was 936 = 2 ? 324 + 288, which gives 288 = (a ? 2b) ? 2(7b ? 3a) = 7a ? 16b. The fourth equation was 324 = 1 ? 288 + 36, Factorization and the Primes which gives 36 = (7b ? 3a) ? (7a ? 16b) = 23b ? 10a. 21 This expresses the highest common factor, 36, as the take issueence of two multiples of the numbers a and b. If one prefers an expression in which the multiple of a comes ? rst, this can be obtained by arguing that 23b ? 10a = (M ? 10)a ? (N ? 23)b, provided that Ma = N b.Since a and b have the common factor 36, this factor can be removed from both of them, and the condition on M and N becomes 200M = 87N . The simplest choice for M and N is M = 87, N = 200, which on reversal gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another form. Suppose a, b, n are disposed(p) natural numbers, and it is coveted to ? nd natural numbers x and y such that ax ? by = n. (6) Such an equation is called an indeterminate equation since it does not determine x and y work outly, or a Diophantine equation after Diophantus of Alexandria (third coulomb A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be alcohol- alcohol-soluble unless n is a multiple of the highest common factor h of a and b for this highest common factor divides ax ? by, whatever set x and y may have. Now suppose that n is a multiple of h, say, n = mh. Then we can solve the equation for all we have to do is ? rst solve the equation ax1 ? by1 = h, as we have seen how to do above, and then multiply throughout by m, getting the solution x = mx1 , y = my1 for the equation (6). Hence the linear indeterminate equation (6) is soluble in natural numbers x, y if and only if n is a multiple of h.In particular, if a and b are comparatively prime, so that h = 1, the equation is soluble whatever value n may have. As regards the linear indeterminate equation ax + by = n, we have found the condition for it to be soluble, not in natural numbers, but in integers of opposite signs one positive and one negative. The questio n of when this equation is soluble in natural numbers is a more dif? cult one, and one that cannot well be completely answered in any simple way. sure enough 22 The Higher Arithmetic n must be a multiple of h, but also n must not be too small in relation to a and b.It can be proved quite slowly that the equation is soluble in natural numbers if n is a multiple of h and n ab. 9. Factorizing a number The obvious way of factorizing a number is to test whether it is divisible by 2 or by 3 or by 5, and so on, using the series of primes. If a number N v is not divisible by any prime up to N , it must be itself a prime for any composite number has at least two prime factors, and they cannot both be v greater than N . The process is a very laborious one if the number is at all large, and for this reason factor tables have been computed.The most extensive one which is generally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909 reprinted by Hafner Press , naked as a jaybird York, 1956), which gives the least prime factor of each number up to 10,000,000. When the least prime factor of a particular number is known, this can be split out, and repetition of the process gives eventually the complete factorization of the number into primes. Several mathematicians, among them Fermat and Gauss, have invented methods for reducing the amount of trial that is necessary to factorize a large number.Most of these involve more knowledge of number-theory than we can consume at this stage but there is one method of Fermat which is in principle extremely simple and can be explained in a few words. Let N be the abandoned number, and let m be the least number for which m 2 N . Form the numbers m 2 ? N , (m + 1)2 ? N , (m + 2)2 ? N , . . . . (7) When one of these is reached which is a perfect square, we get x 2 ? N = y 2 , and consequently N = x 2 ? y 2 = (x ? y)(x + y). The calculation of the numbers (7) is facilitated by noting that their seque nt resistences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily do by using Barlows Table of Squares. The method is particularly successful if the number N has a factorization in which the two factors are of about the same magnitude, since then y is small. If N is itself a prime, the process goes on until we reach the solution provided by x + y = N , x ? y = 1. As an illustration, take N = 9271. This comes between 962 and 972 , so that m = 97. The ? rst number in the series (7) is 972 ? 9271 = 138. The Factorization and the Primes 23 subsequent ones are obtained by adding successively 2m + 1, then 2m + 3, and so on, that is, 195, 197, and so on.This gives the series 138, 333, 530, 729, 930, . . . . The fourth of these is a perfect square, namely 272 , and we get 9271 = 1002 ? 272 = 127 ? 73. An interesting algorithm for factorization has been discovered recently by Captain N. A. Draim, U . S . N. In this, the result of each trial div ision is used to vary the number in preparation for the next division. There are several forms of the algorithm, but perhaps the simplest is that in which the successive divisors are the mirthful numbers 3, 5, 7, 9, . . . , whether prime or not. To explain the rules, we work a numerical example, say N = 4511. The ? st step is to divide by 3, the quotient being 1503 and the remainder 2 4511 = 3 ? 1503 + 2. The next step is to subtract doubly the quotient from the given number, and then add the remainder 4511 ? 2 ? 1503 = 1505, 1505 + 2 = 1507. The last number is the one which is to be carve up by the next odd number, 5 1507 = 5 ? 301 + 2. The next step is to subtract twice the quotient from the ? rst derived number on the previous line (1505 in this case), and then add the remainder from the last line 1505 ? 2 ? 301 = 903, 903 + 2 = 905. This is the number which is to be divided by the next odd number, 7. Now we an continue in just the same way, and no further explanation will be necessitate 905 = 7 ? 129 + 2, 903 ? 2 ? 129 = 645, 645 ? 2 ? 71 = 503, 503 ? 2 ? 46 = 411, 645 + 2 = 647, 503 + 8 = 511, 411 + 5 = 416, 647 = 9 ? 71 + 8, 511 = 11 ? 46 + 5, 416 = 13 ? 32 + 0. 24 The Higher Arithmetic We have reached a zero remainder, and the algorithm tells us that 13 is a factor of the given number 4511. The complementary factor is found by carrying out the ? rst half of the next step 411 ? 2 ? 32 = 347. In fact 4511 = 13? 347, and as 347 is a prime the factorization is complete. To justify the algorithm generally is a matter of mere(a) algebra.Let N1 be the given number the ? rst step was to express N1 as N1 = 3q1 + r1 . The next step was to form the numbers M2 = N1 ? 2q1 , The number N2 was divided by 5 N2 = 5q2 + r2 , and the next step was to form the numbers M3 = M2 ? 2q2 , N 3 = M3 + r 2 , N 2 = M2 + r 1 . and so the process was continued. It can be deduced from these equations that N2 = 2N1 ? 5q1 , N3 = 3N1 ? 7q1 ? 7q2 , N4 = 4N1 ? 9q1 ? 9q2 ? 9q3 , and s o on. Hence N2 is divisible by 5 if and only if 2N1 is divisible by 5, or N1 divisible by 5. Again, N3 is divisible by 7 if and only if 3N1 is divisible by 7, or N1 divisible by 7, and so on.When we reach as divisor the least prime factor of N1 , exact divisibility occurs and there is a zero remainder. The general equation analogous to those given above is Nn = n N1 ? (2n + 1)(q1 + q2 + + qn? 1 ). The general equation for Mn is found to be Mn = N1 ? 2(q1 + q2 + + qn? 1 ). (9) If 2n + 1 is a factor of the given number N1 , then Nn is scarce divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 + + qn ), (8) Factorization and the Primes by (8). below these circumstances, we have, by (9), Mn+1 = N1 ? 2(q1 + q2 + + qn ) = N1 ? 2 n 2n + 1 N1 = N1 . n + 1 25 Thus the complementary factor to the factor 2n + 1 is Mn+1 , as stated in the example. In the numerical example worked out above, the numbers N1 , N2 , . . . decrease steadily. This is always the case at the beginning of the algorithm, but may not be so later. However, it appears that the later numbers are always considerably less than the original number. 10. The series of primes Although the notion of a prime is a very natural and obvious one, questions concerning the primes are often very dif? cult, and galore(postnominal) such questions are quite unanswerable in the present state of mathematical knowledge.We conclude this chapter by mentioning brie? y some results and conjectures about the primes. In 3 we gave Euclids proof that there are in? nitely numerous primes. The same argument will also serve to prove that there are in? nitely umteen primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two betterments (a) 1, 5, 9, 13, 17, 21, 25, . . . , (b) 3, 7, 11, 15, 19, 23, 27, . . . the progression (a) consisting of all numbers of the form 4x + 1, and the progression (b) of all numbers of the form 4x ? 1 (or 4x + 3, which comes to the same thing).We ? rst prove that there are in? nitely umpteen another(prenominal) primes in the progression (b). Let the primes in (b) be enumerated as q1 , q2 , . . . , beginning with q1 = 3. Consider the number N de? ned by N = 4(q1 q2 . . . qn ) ? 1. This is itself a number of the form 4x ? 1. Not every prime factor of N can be of the form 4x + 1, because any product of numbers which are all of the form 4x + 1 is itself of that form, e. g. (4x + 1)(4y + 1) = 4(4x y + x + y) + 1. Hence the number N has some prime factor of the form 4x ? 1. This cannot be any of the primes q1 , q2 , . . . , qn , since N leaves the remainder ? when 26 The Higher Arithmetic divided by any of them. Thus there exists a prime in the series (b) which is several(predicate) from any of q1 , q2 , . . . , qn and this proves the proposition. The same argument cannot be used to prove that there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it d oes not follow that this number will necessarily have a prime factor of that form. However, another argument can be used. Let the primes in the series (a) be enumerated as r1 , r2 , . . . , and consider the number M de? ned by M = (r1 r2 . . rn )2 + 1. We shall see later (III. 3) that any number of the form a 2 + 1 has a prime factor of the form 4x + 1, and is indeed entirely composed of such primes, together possibly with the prime 2. Since M is obviously not divisible by any of the primes r1 , r2 , . . . , rn , it follows as before that there are in? nitely many primes in the progression (a). A similar bureau arises with the two progressions 6x + 1 and 6x ? 1. These progressions exhaust all numbers that are not divisible by 2 or 3, and therefore every prime after 3 falls in one of these two progressions.One can prove by methods similar to those used above that there are in? nitely many primes in each of them. But such methods cannot cope with the general arithmetical progression. Such a progression consists of all numbers ax +b, where a and b are ? xed and x = 0, 1, 2, . . . , that is, the numbers b, b + a, b + 2a, . . . . If a and b have a common factor, every number of the progression has this factor, and so is not a prime (apart from possibly the ? rst number b). We must therefore suppose that a and b are relatively prime. It then seems plausible that the progression will contain in? itely many primes, i. e. that if a and b are relatively prime, there are in? nitely many primes of the form ax + b. Legendre seems to have been the ? rst to realize the importance of this proposition. At one time he thought he had a proof, but this dour out to be fallacious. The ? rst proof was given by Dirichlet in an important memoir which appeared in 1837. This proof used analytical methods (functions of a consecutive variable, limits, and in? nite series), and was the ? rst rightfully important application of such methods to the theory of numbers.It open(a) up com pletely new lines of development the ideas underlying Dirichlets argument are of a very general character and have been fundamental for much subsequent work applying analytical methods to the theory of numbers. Factorization and the Primes 27 Not much is known about other forms which represent in? nitely many primes. It is conjectured, for instance, that there are in? nitely many primes of the form x 2 + 1, the ? rst few being 2, 5, 17, 37, 101, 197, 257, . . . . But not the slightest progress has been made towards proving this, and the question seems hopelessly dif? cult.Dirichlet did succeed, however, in proving that any quadratic form in two variables, that is, any form ax 2 + bx y + cy 2 , in which a, b, c are relatively prime, represents in? nitely many primes. A question which has been deeply investigated in modern times is that of the frequency of event of the primes, in other words the question of how many primes there are among the numbers 1, 2, . . . , X when X is large. This number, which depends of course on X , is normally denoted by ? (X ). The ? rst conjecture about the magnitude of ? (X ) as a function of X seems to have been made independently by Legendre and Gauss about X 1800.It was that ? (X ) is approximately log X . Here log X denotes the natural (so-called Napierian) logarithm of X , that is, the logarithm of X to the base e. The conjecture seems to have been based on numerical evidence. For example, when X is 1,000,000 it is found that ? (1,000,000) = 78,498, whereas the value of X/ log X (to the nearest integer) is 72,382, the ratio being 1. 084 . . . . Numerical evidence of this kind may, of course, be quite misleading. But here the result suggested is true, in the sense that the ratio of ? (X ) to X/ log X tends to the limit 1 as X tends to in? ity. This is the famous Prime Number Theorem, ? rst proved by Hadamard and de la Vall? e e Poussin independently in 1896, by the use of new and powerful analytical methods. It is impossible t o give an account here of the many other results which have been proved concerning the distribution of the primes. Those proved in the nineteenth century were mostly in the nature of imperfect approaches towards the Prime Number Theorem those of the twentieth century included confused re? nements of that theorem. There is one recent event to which, however, reference should be made.We have already said that the proof of Dirichlets Theorem on primes in arithmetical progressions and the proof of the Prime Number Theorem were analytical, and made use of methods which cannot be said to belong properly to the theory of numbers. The propositions themselves relate entirely to the natural numbers, and it seems tenable that they should be provable without the intervention of such foreign ideas. The search for elementary proofs of these two theorems was unsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlets Theorem, and with 28 The Higher Arith metic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An elementary proof, in this connection, authority a proof which operates only with natural numbers. Such a proof is not necessarily simple, and indeed both the proofs in question are distinctly dif? cult. Finally, we may mention the famous problem concerning primes which was propounded by Goldbach in a letter to Euler in 1742. Goldbach suggested (in a slightly different wording) that every even number from 6 onwards is representable as the sum of two primes other than 2, e. g. 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 = 5 + 5, 12 = 5 + 7, . . . Any problem like this which relates to additive properties of primes is necessarily dif? cult, since the de? nition of a prime and the natural properties of primes are all expressed in terms of multiplication. An important donation to the subject was made by Hardy and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that co uld be considered as even a irrelevant approach towards a solution of Goldbachs problem. In that year the Russian mathematician Schnirelmann proved that there is some number N such that every number from some point onwards is representable as the sum of at most N primes.A much nearer approach was made by Vinogradov in 1937. He proved, by analytical methods of extreme subtlety, that every odd number from some point onwards is representable as the sum of three primes. This was the starting point of much new work on the additive theory of primes, in the course of which many problems have been solved which would have been quite beyond the scope of any pre-Vinogradov methods. A recent result in connection with Goldbachs problem is that every suf? ciently large even number is representable as the sum of two numbers, one of which is a prime and the other of which has at most two prime factors.Notes Where material is changing more rapidly than print cycles permit, we have chosen to place s ome of the material on the books website www. cambridge. org/davenport. Symbols such as I0 are used to indicate where there is such additional material. 1. The main dif? culty in giving any account of the laws of arithmetic, such as that given here, lies in deciding which of the various concepts should come ? rst. There are several possible arrangements, and it seems to be a matter of taste which one prefers. It is no part of our purpose to contemplate further the concepts and laws of ? rithmetic. We take the commonsense (or na? ve) view that we all know Factorization and the Primes 29 the natural numbers, and are satis? ed of the validity of the laws of arithmetic and of the principle of induction. The reader who is interested in the foundations of mathematics may mention Bertrand Russell, Introduction to Mathematical Philosophy (Allen and Unwin, London), or M. Black, The Nature of Mathematics (Harcourt, Brace, New York). Russell de? nes the natural numbers by selecting them from numbers of a more general kind. These more general numbers are the (? ite or in? nite) central numbers, which are de? ned by means of the more general notions of class and one-to-one correspondence. The selection is made by de? ning the natural numbers as those which possess all the inductive properties. (Russell, loc. cit. , p. 27). But whether it is fairish to base the theory of the natural numbers on such a vague and dissatisfactory concept as that of a class is a matter of opinion. Dolus latet in universalibus as Dr Johnson remarked. 2. The objection to using the principle of induction as a de? ition of the natural numbers is that it involves references to any proposition about a natural number n. It seems plain the that propositions envisaged here must be statements which are signi? cant when made about natural numbers. It is not clear how this signi? cance can be tested or appreciated except by one who already knows the natural numbers. 4. I am not aware of having seen thi s proof of the uniqueness of prime factorization elsewhere, but it is unlikely that it is new. For other direct proofs, see Mathews, p. 2, or Hardy and Wright, p. 21.? 5. It has been shown by (intelligent computer searches that there is no odd perfect number less than 10300 . If an odd perfect number exists, it has at least eight distinct prime factors, of which the largest exceeds 108 . For references and other development on perfect or nearly perfect numbers, see Guy, sections A. 3, B. 1 and B. 2. I1 6. A critical reader may expose that in two places in this section I have used principles that were not explicitly stated in 1 and 2. In each place, a proof by induction could have been given, but to have done so would have distracted the readers attention from the main issues.The question of the length of Euclids algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuths The Art of Computer Programming vol. II Seminumerical Algorithms (Addison Wesley, Reading, Mass. , 3rd. ed. , 1998) section 4. 5. 3. 9. For an account of early methods of factoring, see Dicksons History Vol. I, ch. 14. For a discussion of the subject as it appeared in ? Particulars of books referred to by their authors names will be found in the Bibliography. 30 The Higher Arithmetic the 1970s see the article by Richard K. Guy, How to factor a number, Congressus Numerantium XVI Proc. th Manitoba Conf. Numer. Math. , Winnipeg, 1975, 4989, and at the turn of the millennium see Richard P. Brent, Recent progress and prospects for integer factorization algorithms, Springer Lecture Notes in Computer Science 1858 Proc. Computing and Combinatorics, 2000, 322. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmers tables will ever be extended, since with them and a pocket calculator one can easily check whether a 12-digit number is a prime. Primality testing is discussed in VIII. 2 and VIII. 9. For Draims algorithm, see Mathematics Magazine, 25 (1952) 1914. 10. An excellent account of the distribution of primes is given by A. E. Ingham, The Distribution of Prime Numbers (Cambridge Tracts, no. 30, 1932 reprinted by Hafner Press, New York, 1971). For a more recent and extensive account see H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). H. Iwaniec (Inventiones Math. 47 (1978) 17188) has shown that for in? nitely many n the number n 2 + 1 is either prime or the product of at most two primes, and indeed the same is true for any irreducible an 2 + bn + c with c odd. Dirichlets proof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dicksons Modern dim-witted Theory of Numbers. An elementary proof of the Prime Number Theorem is given in ch. 22 of Hardy and Wright. An elementary proof of the asymptotic formula for the number of primes in an arithmetic progression is given in Gelfond and Linnik, ch. 3. For a canvas of early work on Goldbachs problem, see James, Bull. American Math. Soc. , 5 5 (1949) 24660. It has been veri? ed that every even number from 6 to 4 ? 1014 is the sum of two primes, see Richstein, Math. Comp. , 70 (2001) 17459. For a proof of Chens theorem that every suf? iently large even integer can be represented as p + P2 , where p is a prime, and P2 is either a prime or the product of two primes, see ch. 11 of Sieve Methods by H. Halberstam and H. E. Richert (Academic Press, London, 1974). For a proof of Vinogradovs result, see T. Estermann, Introduction to Modern Prime Number Theory (Cambridge Tracts, no. 41, 1952) or H. Davenport, Multiplicative Number Theory, 3rd. ed. (Springer, 2000). Suf? ciently large in Vinogradovs result has now been quanti? ed as greater than 2 ? 101346 , see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 1033). e II CONGRUENCES 1. The congruence notation It often happens that for the purposes of a particular cal culation, two numbers which differ by a multiple of some ? xed number are equivalent, in the sense that they produce the same result. For example, the value of (? 1)n depends only on whether n is odd or even, so that two determine of n which differ by a multiple of 2 give the same result. Or again, if we are concerned only with the last digit of a number, then for that purpose two umbers which differ by a multiple of 10 are effectively the same. The congruence notation, introduced by Gauss, serves to express in a convenient form the fact that two integers a and b differ by a multiple of a ? xed natural number m. We say that a is appropriate to b with respect to the modulus m, or, in symbols, a ? b (mod m). The meaning of this, then, is simply that a ? b is divisible by m. The notation facilitates calculations in which numbers differing by a multiple of m are effectively the same, by stressing the parity between congruence and equality.Congruence, in fact, means equality except fo r the addition of some multiple of m. A few examples of valid congruences are 63 ? 0 (mod 3), 7 ? ?1 (mod 8), 52 ? ?1 (mod 13). A congruence to the modulus 1 is always valid, whatever the two numbers may be, since every number is a multiple of 1. Two numbers are appropriate with respect to the modulus 2 if they are of the same parity, that is, both even or both odd. 31 32 The Higher Arithmetic Two congruences can be added, subtracted, or cipher together, in just the same way as two equations, provided all the congruences have the same modulus.If a ? ? (mod m) and b ? ? (mod m) then a + b ? ? + ? (mod m), a ? b ? ? ? ? (mod m), ab ? (mod m). The ? rst two of these statements are immediate for example (a + b) ? (? + ? ) is a multiple of m because a ? ? and b ? ? are both multiples of m. The third is not quite so immediate and is best proved in two steps. First ab ? ?b because ab ? ?b = (a ? ?)b, and a ? ? is a multiple of m. Next, ? b ? , for a similar reason. Hence ab ? (mod m). A congruence can always be multiplied throughout by any integer if a ? ? (mod m) then ka ? k? (mod m).Indeed this is a special case of the third result above, where b and ? are both k. But it is not always legitimate to cancel a factor from a congruence. For example 42 ? 12 (mod 10), but it is not permissible to cancel the factor 6 from the numbers 42 and 12, since this would give the false result 7 ? 2 (mod 10). The reason is obvious the ? rst congruence states that 42 ? 12 is a multiple of 10, but this does not imply that 1 (42 ? 12) is a multiple of 10. The cancellation of 6 a factor from a congruence is legitimate if the factor is relatively prime to the modulus.For let the given congruence be ax ? ay (mod m), where a is the factor to be cancelled, and we suppose that a is relatively prime to m. The congruence states that a(x ? y) is divisible by m, and it follows from the last proposition in I. 5 that x ? y is divisible by m. An illustration of the use of congruences is provid ed by the well-known rules for the divisibility of a number by 3 or 9 or 11. The usual representation of a number n by digits in the scale of 10 is really a representation of n in the form n = a + 10b + 100c + , where a, b, c, . . . re the digits of the number, read from right to left, so that a is the number of units, b the number of tens, and so on. Since 10 ? 1 (mod 9), we have also 102 ? 1 (mod 9), 103 ? 1 (mod 9), and so on. Hence it follows from the above representation of n that n ? a + b + c + (mod 9). Congruences 33 In other words, any number n differs from the sum of its digits by a multiple of 9, and in particular n is divisible by 9 if and only if the sum of its digits is divisible by 9. The same applies with 3 in place of 9 throughout. The rule for 11 is based on the fact that 10 ? ?1 (mod 11), so that 102 ? +1 (mod 11), 103 ? 1 (mod 11), and so on. Hence n ? a ? b + c ? (mod 11). It follows that n is divisible by 11 if and only if a ? b+c? is divisible by 1 1. For example, to test the divisibility of 9581 by 11 we form 1? 8+5? 9, or ? 11. Since this is divisible by 11, so is 9581. 2. Linear congruences It is obvious that every integer is congruous (mod m) to exactly one of the numbers 0, 1, 2, . . . , m ? 1. (1) r m, For we can express the integer in the form qm + r , where 0 and then it is congruent to r (mod m). Obviously there are other sets of numbers, besides the set (1), which have the same property, e. . any integer is congruent (mod 5) to exactly one of the numbers 0, 1, ? 1, 2, ? 2. Any such set of numbers is said to compose a complete set of residues to the modulus m. Another way of expressing the de? nition is to say that a complete set of residues (mod m) is any set of m numbers, no two of which are congruent to one another. A linear congruence, by analogy with a linear equation in elementary algebra, means a congruence of the form ax ? b (mod m). (2) It is an important fact that any such congruence is soluble for x, pro vided that a is relatively prime to m.The simplest way of proving this is to observe that if x runs through the numbers of a complete set of residues, then the corresponding value of ax also constitute a complete set of residues. For there are m of these numbers, and no two of them are congruent, since ax 1 ? ax2 (mod m) would involve x1 ? x2 (mod m), by the cancellation of the factor a (permissible since a is relatively prime to m). Since the numbers ax form a complete set of residues, there will be exactly one of them congruent to the given number b. As an example, consider the congruence 3x ? 5 (mod 11). 34 The Higher ArithmeticIf we give x the value 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the determine 0, 3, 6, . . . , 30. These form another complete set of residues (mod 11), and in fact they are congruent respectively to 0, 3, 6, 9, 1, 4, 7, 10, 2, 5, 8. The value 5 occurs when x = 9, and so x = 9 is a solution of the congruence. Naturally any number congruent to 9 (mod 11) will also satisfy the congruence but nevertheless we say that the congruence has one solution, meaning that there is one solution in any complete set of residues. In other words, all solutions are mutually congruent.The same applies to the general congruence (2) such a congruence (provided a is relatively prime to m) is simply equivalent to the congruence x ? x0 (mod m), where x0 is one particular solution. There is another way of looking at the linear congruence (2). It is equivalent to the equation ax = b + my, or ax ? my = b. We proved in I. 8 that such a linear Diophantine equation is soluble for x and y if a and m are relatively prime, and that fact provides another proof of the solubility of the linear congruence. But the proof given above is simpler, and illustrates the advantages gained by using the congruence notation.The fact that the congruence (2) has a unique solution, in the sense explained above, suggests that one may use this solut ion as an interpretation b for the fraction a to the modulus m. When we do this, we obtain an arithmetic (mod m) in which addition, subtraction and multiplication are always possible, and division is also possible provided that the divisor is relatively prime to m. In this arithmetic there are only a ? nite number of essentially distinct numbers, namely m of them, since two numbers which are mutually congruent (mod m) are handle as the same.If we take the modulus m to be 11, as an illustration, a few examples of arithmetic mod 11 are 5 ? 9 ? ?2. 3 Any relation connecting integers or fractions in the ordinary sense be true when interpreted in this arithmetic. For example, the relation 5 + 7 ? 1, 5 ? 6 ? 8, 1 2 7 + = 2 3 6 becomes (mod 11) 6 + 8 ? 3, because the solution of 2x ? 1 is x ? 6, that of 3x ? 2 is x ? 8, and that of 6x ? 7 is x ? 3. Naturally the interpretation given to a fraction depends on the modulus, for instance 2 ? 8 (mod 11), but 2 ? 3 (mod 7). The 3 3 Congruences 35 nly limitation on such calculations is that just mentioned, namely that the denominator of any fraction must be relatively prime to the modulus. If the modulus is a prime (as in the above examples with 11), the limitation takes the very simple form that the denominator must not be congruent to 0 (mod m), and this is exactly analogous to the limitation in ordinary arithmetic that the denominator must not be equal to 0. We shall return to this point later (7). 3. Fermats theorem The fact that there are only a ? nite number of essentially different numbers in arithmetic to a modulus m means that there are algebraic relations which are satis? d by every number in that arithmetic. There is nothing analogous to these relations in ordinary arithmetic. Suppose we take any number x and consider its powers x, x 2 , x 3 , . . . . Since there are only a ? nite number of possibilities for these to the modulus m, we must eventually come to one which we have met before, say x h ? x k (mod m), w here k h. If x is relatively prime to m, the factor x k can be cancelled, and it follows that x l ? 1 (mod m), where l ? h ? k. Hence every number x which is relatively prime to m satis? es some congruence of this form. The least exponent l for which x l ? (mod m) will be called the order of x to the modulus m. If x is 1, its order is obviously 1. To illustrate the de? nition, let us calculate the orders of a few numbers to the modulus 11. The powers of 2, taken to the modulus 11, are 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, . . . . Each one is twice the preceding one, with 11 or a multiple of 11 subtracted where necessary to make the result less than 11. The ? rst power of 2 which is ? 1 is 210 , and so the order of 2 (mod 11) is 10. As another example, take the powers of 3 3, 9, 5, 4, 1, 3, 9, . . . . The ? rst power of 3 which is ? 1 is 35 , so the order of 3 (mod 11) is 5.It will be found that the order of 4 is again 5, and so also is that of 5. It will be seen that the successive powers of x are periodic when we have reached the ? rst number l for which x l ? 1, then x l+1 ? x and the previous cycle is repeated. It is plain that x n ? 1 (mod m) if and only if n is a multiple of the order of x. In the last example, 3n ? 1 (mod 11) if and only if n is a multiple of 5. This remains valid if n is 0 (since 30 = 1), and it remains valid also for negative exponents, provided 3? n , or 1/3n , is interpreted as a fraction (mod 11) in the way explained in 2. 36 The Higher ArithmeticIn fact, the negative powers of 3 (mod 11) are obtained by prolonging the series backwards, and the table of powers of 3 to the modulus 11 is n = ?3 ? 2 ? 1 0 1 2 3 4 5 6 . . . 9 5 4 1 3 9 5 4 1 3 . 3n ? . . . Fermat discovered that if the modulus is a prime, say p, then every integer x not congruent to 0 satis? es x p? 1 ? 1 (mod p). (3) In view of what we have seen above, this is equivalent to saying that the order of any number is a divisor of p ? 1. The result (3) was mentioned by Fer mat in a letter to Fr? nicle de Bessy of 18 October 1640, in which he e also stated that he had a proof.But as with most of Fermats discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (16461716). He proved that x p ? x (mod p), which is equivalent to (3), by writing x as a sum 1 + 1 + + 1 of x units (assuming x positive), and then expanding (1 + 1 + + 1) p by the multinomial theorem. The terms 1 p + 1 p + + 1 p give x, and the coef? cients of all the other terms are easily proved to be divisible by p. Quite a different proof was given by Ivory in 1806. If x ? 0 (mod p), the integers x, 2x, 3x, . . . , ( p ? )x are congruent (in some order) to the numbers 1, 2, 3, . . . , p ? 1. In fact, each of these sets constitutes a complete set of residues except that 0 has been omitted from each. Since the two sets are congruent, their products are congruent, and so (x)(2x)(3x) . . . (( p ? 1)x) ? (1)(2)(3) . . . ( p ? 1)(mod p). Cancelling the factors 2, 3, . . . , p ? 1, as is permissible, we obtain (3). One merit of this proof is that it can be extended so as to apply to the more general case when the modulus is no longer a prime. The generalization of the result (3) to any modulus was ? rst given by Euler in 1760.To formulate it, we must begin by considering how many numbers in the set 0, 1, 2, . . . , m ? 1 are relatively prime to m. Denote this number by ? (m). When m is a prime, all the numbers in the set except 0 are relatively prime to m, so that ? ( p) = p ? 1 for any prime p. Eulers generalization of Fermats theorem is that for any modulus m, x ? (m) ? 1 (mod m), provided only that x is relatively prime to m. (4) Congruences 37 To prove this, it is only necessary to modify Ivorys method by omitting from the numbers 0, 1, . . . , m ? 1 not only the number 0, but all numbers which are not relatively prime to m.There remain ? (m) numbers, say a 1 , a2 , . . . , a? , Then the numbers a1 x, a2 x, . . . , a? x are congruent, in some order, to the previous numbers, and on multiplying and cancelling a1 , a2 , . . . , a? (as is permissible) we obtain x ? ? 1 (mod m), which is (4). To illustrate this proof, take m = 20. The numbers less than 20 and relatively prime to 20 are 1, 3, 7, 9, 11, 13, 17, 19, so that ? (20) = 8. If we multiply these by any number x which is relatively prime to 20, the new numbers are congruent to the original numbers in some other order.For example, if x is 3, the new numbers are congruent respectively to 3, 9, 1, 7, 13, 19, 11, 17 (mod 20) and the argument proves that 38 ? 1 (mod 20). In fact, 38 = 6561. where ? = ? (m). 4. Eulers function ? (m) As we have just seen, this is the number of numbers up to m that are relatively prime to m. It is natural to ask what relation ? (m) bears to m. We saw that ? ( p) = p ? 1 for any prime p. It is also easy to evaluate ? ( p a ) for any prime power pa . The only numbers in the set 0, 1, 2, . . . , pa ? 1 which ar e not relatively prime to p are those that are divisible by p. These are the numbers pt, where t = 0, 1, . . , pa? 1 ? 1. The number of them is pa? 1 , and when we subtract this from the total number pa , we obtain ? ( pa ) = pa ? pa? 1 = pa? 1 ( p ? 1). (5) The determination of ? (m) for general values of m is effected by proving that this function is multiplicative. By this is meant that if a and b are any two relatively prime numbers, then ? (ab) = ? (a)? (b). (6) 38 The Higher Arithmetic To prove this, we begin by law-abiding a general principle if a and b are relatively prime, then two simultaneous congruences of the form x ? ? (mod a), x ? ? (mod b) (7) are precisely equivalent to one congruence to the modulus ab.For the ? rst congruence means that x = ? + at where t is an integer. This satis? es the second congruence if and only if ? + at ? ? (mod b), or at ? ? ? ? (mod b). This, being a linear congruence for t, is soluble. Hence the two congruences (7) are simultaneously so luble. If x and x are two solutions, we have x ? x (mod a) and x ? x (mod b), and therefore x ? x (mod ab). Thus there is exactly one solution to the modulus ab. This principle, which extends at once to several congruences, provided that the moduli are relatively prime in pairs, is sometimes called the Chinese remainder theorem.It assures us of the existence of numbers which leave plus remainders on division by the moduli in question. Let us represent the solution of the two congruences (7) by x ? ? , ? (mod ab), so that ? , ? is a certain number depending on ? and ? (and also on a and b of course) which is unambiguously determined to the modulus ab. Different pairs of values of ? and ? give rise to different values for ? , ? . If we give ? the values 0, 1, . . . , a ? 1 (forming a complete set of residues to the modulus a) and similarly give ? the values 0, 1, . . . , b ? 1, the resulting values of ? , ? constitute a complete set of residues to the modulus ab. It is obvious tha t if ? has a factor in common with a, then x in (7) will also have that factor in common with a, in other words, ? , ? will have that factor in common with a. Thus ? , ? will only be relatively prime to ab if ? is relatively prime to a and ? is relatively prime to b, and conversely these conditions will ensure that ? , ? is relatively prime to ab. It follows that if we give ? the ? (a) possible values that are less than a and prime to a, and give ? the ? (b) values that are less than b and prime to b, there result ? (a)? (b) values of ? ? , and these comprise all the numbers that are less than ab and relatively prime to ab. Hence ? (ab) = ? (a)? (b), as asserted in (6). To illustrate the situation arising in the above proof, we tabulate below the values of ? , ? when a = 5 and b = 8. The possible values for ? are 0, 1, 2, 3, 4, and the possible values for ? are 0, 1, 2, 3, 4, 5, 6, 7. Of these there are four values of ? which are relatively prime to a, corresponding to the fact that ? (5) = 4, and four values of ? that are relatively prime to b, Congruences 39 corresponding to the fact that ? (8) = 4, in accordance with the formula (5).These values are italicized, as also are the corresponding values of ? , ? . The last mentioned constitute the sixteen numbers that are relatively prime to 40 and less than 40, thus verifying that ? (40) = ? (5)? (8) = 4 ? 4 = 16. ? ? 0 1 2 3 4 0 0 16 32 8 24 1 25 1 17 33 9 2 10 26 2 18 34 3 35 11 27 3 19 4 20 36 12 28 4 5 5 21 37 13 29 6 30 6 22 38 14 7 15 31 7 23 39 We now return to the original question, that of evaluating ? (m) for any number m. Suppose the factorization of m into prime powers is m = pa q b . . . . Then it follows from (5) and (6) that ? (m) = ( pa ? pa? 1 )(q b ? q b? 1 ) . . . or, more elegantly, ? (m) = m 1 ? For example, ? (40) = 40 1 ? and ? (60) = 60 1 ? 1 2 1 2 1 p 1? 1 q . (8) 1? 1 3 1 5 = 16, 1 5 1? 1? = 16. The function ? (m) has a remarkable property, ? rst given by Gauss in his Disquisitione s. It is that the sum of the numbers ? (d), extended over all the divisors d of a number m, is equal to m itself. For example, if m = 12, the divisors are 1, 2, 3, 4, 6, 12, and we have ? (1) + ? (2) + ? (3) + ? (4) + ? (6) + ? (12) = 1 + 1 + 2 + 2 + 2 + 4 = 12. A general proof can be based either on (8), or directly on the de? nition of the function. 40 The Higher ArithmeticWe have already referred (I. 5) to a table of the values of ? (m) for m 10, 000. The same intensiveness contains a table giving those numbers m for which ? (m) assumes a given value up to 2,500. This table shows that, up to that point at least, every value assumed by ? (m) is assumed at least twice. It seems reasonable to conjecture that this is true generally, in other words that for any number m there is another number m such that ? (m ) = ? (m). This has never been proved, and any attempt at a general proof seems to support with formidable dif? culties. For some special types of numbers the result is easy, e. g. f m is odd, then ? (m) = ? (2m) or again if m is not divisible by 2 or 3 we have ? (3m) = ? (4m) = ? (6m). 5. Wilsons theorem This theorem was ? rst publis