Barriers Of Entering A Foreign Market

Going abroad with our business has been the talk of the globalization age. In this global society, there are growing reasons of why we should expand our business to foreign markets. First, companies are like continuously growing organisms. It cannot exist without the search of growth or of potentials of growth.This is why mangers cannot afford to live in the illusion that their local markets will be sufficient to sustain the need for continuous growth (Khan, 2005). Second, having an established business overseas will strengthen companies’ financial safety significantly by offsetting domestic seasonal fluctuations.Third, expanding to foreign markets is an excellent choice for enhancing companies’ market shares. Fourth, with the extensive promotion of globalization and US’ effort to combat trade protections, there are significantly more enhanced facilities to support foreign investments today compare to a decade ago. In short, entering foreign markets is an importa nt and contemporary discussion subject (Zacharakis, 1996). However, managers have also realized that the decision to internationalize market shares contains considerable amount of risks and barriers.Some of the most recognizable barriers are cultural and language barriers, environmental issues, political issues, etc. In this paper, we are detailing those barriers and providing case examples to strengthen the arguments. II. Barriers of Entering a Foreign Market II. 1. Cultural and Language Barriers In this discussion, we will start with what is probably the strongest factor that influences expansion to foreign markets. Managers have long accepted that in internationalization considerations, differences between home culture and the culture of foreign countries are significant.Culture is a complex term. It consists of various factors like languages, religions, social norms etc. Thus, companies generally spend considerable portion of their time learning about the culture of the foreign target markets. This is also true whether managers decided to establish new firms in foreign markets or collaborating with foreign partners. Studies also indicated that cultural issues influence the manner in which companies perform their international expansion. Firms generally increase their commitment in investing to a particular foreign target market in predictable stages.First, they will use export agents to learn about the country’s culture. This type of foreign investment will change along with time and enhanced knowledge about local culture of the target market. II. 2. Business Environment Barriers The local business environment has also been an influential factor that strongly affects foreign expansion activities. For instance, companies can have the problem of not having the sufficient good image in a society that has local preferences. Reputation is the issue resulted from the local business environment condition of several markets with local preferences.Some consu mers have more confidence or tendency to purchase local products rather than foreign made. Despite the extensive marketing efforts performed by foreign companies to take away local market share, they still lagged behind local products, even ones with less marketing budget. II. 3. Political and Government Regulations Barriers Other barriers are political in nature. Governmental policies can create enormous effect on company’s success or failure in entering foreign markets. China is the most apparent example of this premise.The Chinese markets have been closed from foreign investors for decades before a massive governmental revolution created opportunities for foreign investment. The government opens chances for FDI inflow. Furthermore, supports foreign investment by means of incentives, property rights protections, etc. Afterwards, economic records indicated that the country has been experiencing one of the most rapid growths in the world, with an average annual GDP growth per centage of 10% for the last decade. In short, governmental policies have significant importance in international expansion.III. Several Cases from 2001-2006 In this paper, I will provide several examples of cases involving foreign entry barriers mentioned above. Despite the similar nature of barriers in each cases, each country has their own tendency of foreign trade barriers. III. 1. Entering Indonesian Markets Indonesian is seen as one of the most economically potential markets in Asia today. Its abundant amount of human resources and cheap labor has been considerable attractions for international investors since the country recovered from its economic crisis.Nevertheless, the country is recorded to have several issues that might hamper international investment toward local markets. First, in terms of governmental policies, the country is still enacting several import and export restrictions to protect local consumers and to ensure that local necessities are fulfilled before forei gn investors could take a share of the market. This could mean higher tariffs, longer bureaucracy, etc. Second, the country has a unique set of culture.Cultural analysts and foreign managers operating in local markets described the country as being comfortable in doing things their own way and refuse to have it challenged (Forrest, 2001). The importance of physical presence of superiors, the lack appreciation toward punctuality and the respect for age and seniority is several of many things that must be learned about Indonesian culture before entering local markets. Learning informal business etiquettes are often as important as learning formal ones, or sometimes more important.For example, there is a significant cultural practice in Indonesia when commonly, Indonesian managers tend to hire their relatives and friends regardless their competences. This situation is inappropriate for Australian or American companies since they consider it as nepotism (Dowling & De Cieri, 1989). III. 2. Japanese Firms Entering US Markets In the case of Japan companies’ expansion to US markets, the case lies in condition of US’ business environment. Most US consumers prefer national products rather than foreign ones. This creates significant challenges for Japanese companies targeting US markets.Some Japan companies perform large marketing effort to facilitate their presence in US local markets. However, as mentioned previously, some of these efforts did not work as planned. Locals could still easily take control of the market share. This is identified as the barrier of reputation. The study of Japanese companies who enters US market revealed that some Japan companies chose collaboration with local brands in order to win local preferences rather than performing endless marketing campaigns that could have weak effects (Chen, 2003).Concerning the decision making, for example, Japanese managers tend explore the roots of problem before making a particular decision. In c ontrast, American managers are likely to adopt straightforward approach (judgmental behavior) that is much efficient than Japanese approach but less effective. Following link, inform the practice of Japanese culture in terms of big typhoon etc (http://www. brovision. com/) and http://www. mccombs. utexas. edu/research/ciber/executivevideotapes. asp. sssIn foreign countries, for instances, Japanese companies like Toyota and Honda that realize their HR practices are unacceptable by non-Japanese culture may come up with an unfortunate solution by hiring employees under distinct employment categories that lack of job security (Hersey, 1972). III. 3. United States and China In the recent case of United State’s commerce department and the government of China, another foreign trade issues caused by local business environment appear. US Department of Commerce’s assistant secretary stated that China has been using technical regulations as a barrier of trade barriers.This is don e by imposing certain quality standards that would effectively band certain products from entering the Chinese local markets. US department of commerce are currently fighting to oppose this type of trade barriers using diplomatic means (‘United States’, 2005). Bibliography Chen, Shih-Fen. Zeng Ming. 2003. ‘Japanese Investor’s Choice of Acquisition vs Startup in the US: The Role of Reputation Barriers and Advertising Outlays’. International Journal of Research in Marketing. Retrieved February 14, 2007 from brandeis. edu/ibs/faculty_publications/chen/japanese_acquisitions.pdf Dowling, P. J. , Welch, D. E. & De Cieri, H. 1989, ‘International joint ventures: a new challenge for human management’, Proceedings of the fifteenth conference of the European international business association. Helsinki, December, 1989 Forrest, W. , Bidgood, M. 2001. Cultural Aspects of Business. American Indonesia Chamber of Commerce. www. aiccusa. org Fiedler, Fre d E. 1965. Engineer the Job to Fit the Manager. Harvard Business Review. Vol. 43 Hersey, Paul. Blanchard, Kenneth H. 1972. Management of Organization Behavior. New Jersey: Prentic- Hall Inc. Kenna, Peggy.Sondra, Lacy. 1994. Business Japan: A Practical Guide to Understanding Japanese Business Culture. McGraw-Hill Khan, Asim. 2005. Business Management Inc. Retrieved February 14, 2007 from www. themanager. org/strategy/Deciding_to_Go_International. pdf ‘United States Combating Use of Standards as Trade Barriers’. 2005. US INFO. STATE. GOV. Retrieved February 14, 2007 from http://usinfo. state. gov/xarchives/display. html? p=washfile-english&y=2005&m=May&x=20050513162339ajesroM0. 5901605&t=livefeeds/wf-latest. html Zacharakis, Andrew. 1996. Academy of Management Executive. 10(4): 109-110.

The Higher Arithmetic – an Introduction to the Theory of Numbers

This page intentionally left blank Now into its eighth edition and with additional material on primality testing, written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signi? cance. A companion website (www. cambridge. org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . can be recommended both for independent study and as a reference text for a general mathematical audience. ’ European Maths Society Journal ‘Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer’s opinion, is far superior for this purpose to any other book in English. ’ Bulletin of the American Mathematical Society THE HIGHER ARITHMETIC AN INTRODUCTION TO THE THEORY OF NUMBERS Eighth edition H. Davenport M. A. , SC. D. F. R. S. late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by James H. Davenport CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www. cambridge. org Information on this title: www. cambridge. org/9780521722360  © The estate of H. Davenport 2008 This publication is in copyright.Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission 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 not guarantee that any content on such websites is, or will remain, accurate or appropriate. CONTENTS Introduction I Factorization and the Primes 1. 2. 3. 4. . 6. 7. 8. 9. 10. The laws of arithmetic Proof by induction Prime numbers The fundamental theorem of arithmetic Consequences of the fundamental theorem Euclid’s algorithm Another proof of the fundamental theorem A property of the H. C. F Factorizing a number The series of primes page 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 congruence notation Linear congruences Fermat’s theorem Euler’s function ? (m) Wilson’s theorem Algebraic congruences Congruences to a prime modulus Congr uences in several unknowns Congruences covering all numbers v vi III Quadratic Residues 1. 2. 3. 4. . 6. Primitive roots Indices Quadratic residues Gauss’s lemma The law of reciprocity The 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 Continued Fractions 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Introduction The general continued fraction Euler’s rule The convergents to a continued fraction The equation ax ? by = 1 In? nite continued fractions Diophantine approximation Quadratic irrationals Purely periodic continued fractions Lagrange’s theorem Pell’s equation A geometrical interpretation of continued fractionsV Sums of Squares 1. 2. 3. 4. 5. Numbers representable by two squares Primes of the form 4k + 1 Constructions for x and y Representation by four squares Representation by three squares VI Quadratic Forms 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Equivalent forms The discriminant The representation of a number by a form Three examples The reduction of positive de? nite forms The reduced forms The number of representations The class-number Contents VII Some Diophantine Equations 1. Introduction 2. The equation x 2 + y 2 = z 2 3. The equation ax 2 + by 2 = z 2 4. Elliptic equations and curves 5.Elliptic equations modulo primes 6. Fermat’s Last Theorem 7. The equation x 3 + y 3 = z 3 + w 3 8. Further developments vii 137 137 138 140 145 151 154 157 159 165 165 168 173 179 185 188 194 199 200 209 222 225 235 237 VIII Computers and Number Theory 1. 2. 3. 4. 5. 6. 7. 8. 9. Introduction Testing for primality ‘Random’ number generators Pollard’s factoring methods Factoring and primality via elliptic curves Factoring large numbers The Dif? e–Hellman cryptographic method The RSA cryptographic method Primality testing revisited Exercises Hints Answers Bibliography IndexINTRODUCTION T he higher arithmetic, or the theory of numbers, is concerned with the properties of the natural numbers 1, 2, 3, . . . . These numbers must have exercised human curiosity from a very early period; and in all the records of ancient civilizations there is evidence of some preoccupation with arithmetic over and above the needs of everyday life. But as a systematic and independent science, the higher arithmetic is entirely a creation of modern times, and can be said to date from the discoveries of Fermat (1601–1665).A peculiarity of the higher arithmetic is the great dif? culty which has often been experienced in proving simple general theorems which had been suggested quite naturally by numerical evidence. ‘It is just this,’ said Gauss, ‘which gives the higher arithmetic that magical charm which has made it the favourite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics. ’ The theory of numbers is generally considered to be the ‘purest’ branch of pure mathematics.It certainly has very few direct applications to other sciences, but it has one feature in common with them, namely the inspiration which it derives from experiment, which takes the form of testing possible general theorems by numerical examples. Such experiment, though necessary in some form to progress in every part of mathematics, has played a greater part in the development of the theory of numbers than elsewhere; for in other branches of mathematics the evidence found in this way is too often fragmentary and misleading.As regards the present book, the author is well aware that it will not be read without effort by those who are not, in some sense at least, mathematicians. But the dif? culty is partly that of the subject itself. It cannot be evaded by using imperfect analogies, or by presenting the proofs in a way viii Introduction ix which may convey the main idea o f the argument, but is inaccurate in detail. The theory of numbers is by its nature the most exact of all the sciences, and demands exactness of thought and exposition from its devotees. The theorems and their proofs are often illustrated by numerical examples.These are generally of a very simple kind, and may be despised by those who enjoy numerical calculation. But the function of these examples is solely to illustrate the general theory, and the question of how arithmetical calculations can most effectively be carried out is beyond the scope of this book. The author is indebted to many friends, and most of all to Professor o Erd? s, Professor Mordell and Professor Rogers, for suggestions and corrections. He is also indebted to Captain Draim for permission to include an account of his algorithm.The material for the ? fth edition was prepared by Professor D. J. Lewis and Dr J. H. Davenport. The problems and answers are based on the suggestions of Professor R. K. Guy. Chapter VIII a nd 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’ proof of Fermat’s Last Theorem, and is grateful to Professor J. H. Silverman for his comments. For the eighth edition, many people contributed suggestions, notably Dr J. F. McKee and Dr G. K. Sankaran.Cambridge University Press kindly re-typeset 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, are 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 natural numbers 1, 2, 3, . . . of ordinary arithmetic. Examples of such propositions are the fundamental theorem (I. 4)? hat every nat ural number can be factorized into prime numbers in one and only one way, and Lagrange’s theorem (V. 4) that every natural number can be expressed as a sum of four or fewer perfect squares. We are not concerned with numerical calculations, except as illustrative examples, nor are we much concerned with numerical curiosities except where they are relevant to general propositions. We learn arithmetic experimentally in early childhood by playing with objects such as beads or marbles. We ? rst learn addition by combining two sets of objects into a single set, and later we learn multiplication, in the form of repeated addition.Gradually we learn how to calculate with numbers, and we become familiar with the laws of arithmetic: laws which probably carry more conviction to our minds than any 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 have seen them form ulated in general terms. They can be expressed as follows. ? References in this form are to chapters and sections of chapters of this book. 1 2 The Higher Arithmetic Addition.Any two natural numbers a and b have a sum, denoted by a + b, which is itself a natural number. The operation of addition satis? es the two laws: a+b =b+a (commutative law of addition), (associative law of addition), a + (b + c) = (a + b) + c the brackets in the last formula serving to indicate the way in which the operations are carried out. Multiplication. Any two natural numbers a and b have a product, denoted by a ? b or ab, which is itself a natural number. The operation of multiplication satis? es the two laws ab = ba a(bc) = (ab)c (commutative law of multiplication), (associative law of multiplication).There is also 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 any two natural numbers, then either a is equal to b o r a is less than b or b is less than a, and of these three possibilities exactly one must occur. The statement that a is less than b is expressed symbolically by a < b, and when this is the case we also say 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 common divisors of a and b.If a is divisible by b, then the common divisors of a and b consist simply of all divisors of b, and there is no more to be said. If a is not divisible by b, we can express a as a multiple of b together with a remainder less than b, that is a = qb + c, where c < b. (2) This is the process of ‘division with a remainder’, and expresses the fact that a, not being a multiple of b, must occur somewhere between two consecutive multiples 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.Moreo ver, 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 same 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 essence of the algorithm lies in the repetition of this argument. If b is divisible by c, the common divisors of b and c consist 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 only 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. Factorization and the Primes 17 Let us suppose, for the sake of de? niteness, that the process terminates when we reach the number h, which is a divisor of the preceding number g.Then the last two equations of the series (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 being the last remainder in Euclid’s algorithm before exact divisibility occurs, i. e. the last non-zero remainder. We have therefore proved that the common divisors of two given 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-zero remainder when Euclid’s 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, 3 132 = 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 steps could be replaced by 936 = 3 ? 324 ? 6, 324 = 9 ? 36. The reason why it is permissible to use negative remainders is that the argument 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 relatively prime? if they have no common divisor except 1, or in other words if their H. C. F. is 1. This will be the case if and only if the last remainder, when Euclid’s 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 independent of that given previously. 8 7. Another proof of t he fundamental theorem The Higher Arithmetic We shall now use Euclid’s algorithm to give another proof of the fundamental theorem of arithmetic, independent of that given in  §4. We begin with a very simple remark, which may be thought to be too obvious to be worth making. Let a, b, n be any natural numbers. The highest common factor 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 Euclid’s algorithm or the fundamental theorem of arithmetic.In fact the result follows at once from Euclid’s 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 multiplied throughout by n. Finally, the last remainder, giving the H. C. F. of na and nb, is nh, wher e h is the H. C. F. of a and b. We apply this simple fact to prove the following theorem, often called Euclid’s theorem, since it occurs as Prop. 30 of Book VII.If a prime divides the product of two numbers, it must divide one of the numbers (or possibly both of them). Suppose the prime p divides the product na of two numbers, and does not divide a. The only factors of p are 1 and p, and therefore the only common factor of p and a is 1. Hence, by the theorem just proved, the H. C. F. of np and na is n. Now p divides np obviously, and divides na by hypothesis. Hence p is a common factor of np and na, and so is a factor of n, since we know that every common factor of two numbers is necessarily a factor of their H. C. F.We have therefore proved that if p divides na, and does not divide a, it must divide n; and this is Euclid’s 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 either 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 cancel 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 alternative 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 Euclid’s algorithm) rather than on a special device such as that used in  §4. On the other hand, it is longer and less direct. 8. A property of the H. C.F From Euclid’s 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 difference between a multiple of a and a multiple 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 values of x and y for which ax ? y is actually equal to h. Before 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 represented 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 natura l 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 number is linearly 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 difference of two numbers: to see this, write the second number as by2 ? ax2 , in accordance 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 Euclid’s algorithm, in the light of this concept. 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 next 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 3 132 = 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 difference 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 substitution gives 36 = 77a ? 177b. Returning to the general theory, we can express the result in another form. Suppose a, b, n are given natural numbers, and it is desired to ? nd natural numbers x and y such that ax ? by = n. (6) Such an e quation is called an indeterminate equation since it does not determine x and y completely, or a Diophantine equation after Diophantus of Alexandria (third century A . D . , who wrote a famous treatise on arithmetic. The equation (6) cannot be 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 values 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 relatively 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 question 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. Certainly 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 easily 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 gener ally accessible is that of D. N. Lehmer (Carnegie Institute, Washington, Pub. No. 105. 1909; reprinted by Hafner Press, New 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 divided 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 postulate 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 given 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 successive differences increase at a constant rate. The identi? ation of one of them as a perfect square is most easily made by using Barlow’s 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 fact orization has been discovered recently by Captain N. A. Draim, U . S . N. In this, the result of each trial division is used to modify 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 odd 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 twice 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 divided 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 divi ded by the next odd number, 7. Now we an continue in exactly the same way, and no further explanation will be needed: 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 elementary 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 so 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 exactly divisible by 2n + 1, and Nn = (2n + 1)qn , whence n N1 = (2n + 1)(q1 + q2 +  ·  ·  · + qn ), (8) Factorization and the Primes by (8). Under 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 many 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 Euclid’s proof that there are in? nitely many primes. The same argument will also serve to prove that there are in? nitely many primes of certain speci? ed forms. Since every prime after 2 is odd, each of them falls into one of the two progressions (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 many 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 different from any of q1 , q2 , . . . , qn ; and this proves the proposition. The same argument cannot be used to prove t hat there are in? nitely many primes in the series (a), because if we construct a number of the form 4x +1 it does 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 situation 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 ar e 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 turned 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 continuous variable, limits, and in? nite series), an d was the ? rst really important application of such methods to the theory of numbers.It opened up completely new lines of development; the ideas underlying Dirichlet’s 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 occurrence of the p rimes, 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 usually 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 to 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 various 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 Dirichlet’s 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 reasonable that they should be provable without the intervention of such foreign ideas. The search for ‘elementary’ proofs of these two theorems was u nsuccessful until fairly recently. In 1948 A. Selberg found the ? rst elementary proof of Dirichlet’s Theorem, and with 28 The Higher Arithmetic he help of P. Erd? s he found the ? rst elementary proof of the Prime Numo ber Theorem. An ‘elementary’ proof, in this connection, means 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 multiplic ation. An important contribution to the subject was made by Hardy and Littlewood in 1923, but it was not until 1930 that anything was rigorously proved that could be considered as even a remote approach towards a solution of Goldbach’s 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 Goldbach’s 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 some of the material on the book’s website: www. cambridge. org/davenport. Symbols such as  ¦I:0 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 analyse 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 consult Bertrand Russe ll, 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) cardinal 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 reasonable to base the theory of the natural numbers on such a vague and unsatisfactory 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 th at ‘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 this 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 information on perfect or ‘nearly perfect’ numbers, see Guy, sections A. 3, B. 1 and B. 2.  ¦I:1  §6. A critical reader may notice that in two places in this section I have used principles that were not explicitly stated in  §Ã‚ §1 and 2. In each place, a proof by induction co uld have been given, but to have done so would have distracted the reader’s attention from the main issues.The question of the length of Euclid’s algorithm is discussed in Uspensky and Heaslet, ch. 3, and D. E. Knuth’s 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 Dickson’s 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, 49–89, and at the turn of the millennium see Richard P. Brent, ‘Recent progress and prospects for integer factorisation algorithms’, Springer Lecture Notes in Comp uter Science 1858 Proc. Computing and Combinatorics, 2000, 3–22. The subject is discussed further in Chapter VIII. It is doubtful whether D. N. Lehmer’s 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 Draim’s algorithm, see Mathematics Magazine, 25 (1952) 191–4. 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) 171–88) 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. Dirichlet’s p roof of his theorem (with a modi? ation due to Mertens) is given as an appendix to Dickson’s Modern Elementary 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 survey of early work on Goldbach’s problem, see James, Bull. American Math. Soc. , 55 (1949) 246–60. 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) 1745–9. For a proof of Chen’s 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 Vinogradov’s 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 Vinogradov’s result has now been quanti? ed as ‘greater than 2 ? 101346 ’, see M. -C. Liu and T. Wang, Acta Arith. , 105 (2002) 133–175.Conversely, we know that it is true up to 1. 13256 ? 1022 (Ramar? and Saouter in J. Number Theory 98 (2003) 10–33). e II CONGRUENCES 1. The congruence notation It often happens that for the purposes of a particular calculation, 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 values 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 congruent 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 analogy between congruence and equality.Congruence, in fact, means ‘equality except for 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 congruent 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 m ultiplied 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 provided 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 11. 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 congruent (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 othe r 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 constitute 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, provided 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 values 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 values 0, 1, 2, . . . , 10 (a complete set of residues to the modulus 11), 3x takes the values 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 precisely equivalent to the con gruence 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 solution 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 w hich are mutually congruent (mod m) are treated 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 remains 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 ordina ry arithmetic that the denominator must not be equal to 0. We shall return to this point later ( §7). 3. Fermat’s 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), where 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 Fermat 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 Fermat’s discoveries, the proof was not published or preserved. The ? rst known proof seems to have been given by Leibniz (1646–1716). He proved that x p ? x (mod p), which is equivalent to (3), b y 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 b y 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. Euler’s generalization of Fermat’s 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 Ivory’s 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 relati vely 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. Euler’s 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 are 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 observing 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 soluble. 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 ar e relatively prime in pairs, is sometimes called ‘the Chinese remainder theorem’.It assures us of the existence of numbers which leave prescribed 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 uniquely 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 that 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 latter 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 Disquisitiones. 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 volume 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 meet 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. Wilson’s theorem This theorem was ? rst publis

A case study on suffering with depression

A case study on suffering with depression Depression is a whole body illness, meaning it affects your body, mood and thoughts. It can be a very serious illness which affects the way you eat and sleep, the way you feel about yourself and the way you think about things. It is more than just a passing mood, and is very different from the usual feelings of sadness and feeling fed up. The feelings of depression usually last more than a few days; they can last for months or even years. If left untreated, these feelings can interfere with the daily life of the individual and can also have an effect on the people around them. Depression can affect anyone at any age, including children, although it is more likely to occur if there has been a family history of depression. Health professionals use different terms to describe depression, these are: depression, depressive illness and clinical depression (NHS, 2009). It is commonly thought that depression is not a real illness. It is seen more like a weakness or a failure in an individua l however, just because it is not visible does not mean it is not real. Types of depression There many different forms of depression, these can range from mild depression through to severe depression and individuals who suffer with severe depression may also show psychotic symptoms. Major depression, probably the most common form of depression, is manifested by a combination of symptoms that interfere with the individual’s ability to eat, sleep, work and study. Usually the individual will lose interest in once pleasurable activities and also has a feeling of hopelessness. Some individuals only have a single depressive episode, while others have recurring episodes. (Psychology Information Online, 2009) Dysthymia is a mild, chronic state of depression and the symptoms are similar to major depression, but less severe. A person may suffer from dysthymia depression for years before being diagnosed, thus they would still continue with everyday life and may not even realise that the y are suffering with depression, they could just have a feeling that something is not quite right. (Psychology Information Online, 2009) Atypical depression is different to major depression in the way that an individual will feel better temporarily when a positive life event occurs, whereas an individual suffering from major depression will nearly always feel low. This type of depression can last for a couple of months or can be with an individual for their entire life. (Depression About.com, 2009) Bipolar disorder, or manic depressive disorder, is an emotional disorder â€Å"in which an individual alternates between states of deep depression and extreme elation.† (Bipolar About.com, 2009) It is characterised by sudden changes in mood, thoughts and behaviour and there is a high suicide rate seen in individuals who suffer from manic depression. The two extremes of depression are where the individual feels very low and mania where the individual feels very high. (NHS, 2009) Pos tpartum depression affects woman, almost always, immediately after childbirth. It is thought that postpartum depression is triggered by the hormonal changes that follow childbirth. Some woman have severe and long lasting symptoms that require treatment, others can generally beat the baby blues with good self-care and support from friends and family. (Depression About.com, 2009)

Marketing Issue Concerning Positioning Strategy of Clean Edge Essay - 37

Marketing Issue Concerning Positioning Strategy of Clean Edge - Essay Example Considering this problem, Randall, the one given the authority to take charge of positioning strategy of Clean Edge has to find alternative courses of actions then analyze them well and choose the probable best among them. The entire problem highlights the idea of marketing issue concerning positioning strategy of Clean Edge. It talks about the point whether it is good to pursue its launching in the market knowing that it could somehow compete with company’s bread and butter such as Paramount Pro and Avail. The company might be a bit hesitant to pursue this issue as there are required data needed to be unearthed prior to the actual decision-making process. Thus, it is not enough to rely on the point that the customers are becoming sophisticated, knowing what they exactly need in the advent of technological advancement. For this reason, Randall, on behalf of the Paramount should consider investigating alternative courses of actions. One alternative course of action is to launch Clean Edge as a niche product within a year or two, which has to be the indicated period of the market trial. For this reason, he has to consider conducting economic analysis in order to be sure about the output. This part icularly would take into account some probable impacts such as the cannibalization effect. So a profit-loss pro forma is necessary in order to find out Clean Edge’s performance and its potential impact on the current profit Paramount’s top products are generating. It is not easy to decide right away. What Randall requires is a thorough market research on this issue taking into account the associated economic analysis and a benefit-loss analysis. In order for Paramount to grow more, it has to significantly analyze the product life cycle of its top products and the extent to which customers are becoming sophisticated with the latest innovations in technology.  

Why Globalization Is Bad for the Economy Research Paper

Why Globalization Is Bad for the Economy - Research Paper Example Macro economics consists of concepts that can be applied to the entire world. Globalization is a procedure of interaction and integration among the people, companies, and governments of different nations, a process driven by international trade and investment and aided by information technology. This process has effects on the environment, on culture, on political systems, on economic development and prosperity, and on human physical well-being in societies around the world. For thousands of years, people and corporations have been buying from and selling to each other in lands at great distances hence globalization is not the new concept, but there have been vast changes in form of technology, advancement and policies over decades. Globalization can be explained as Covering a wide range of distinct political, economic, and cultural trends and discouraging barriers. Globalization became commonplace in the last two decades, and In today’s world barriers and distance don’ t matter anymore especially because of the advancement in technology, media and internal and most of all the mode of travelling. By the help of all the advancement and progress in the technology now anyone can travel a thousand miles in matter of hours and days. However Globalization is a very controversial topic, many economist don’t support the idea of globalization as it has many diverse affects on the economy. As many economist believe that Advances in communication and transportation technology, combined with free-market ideology, have given goods, services, and capital unprecedented mobility. And this can affect the local market of the country. Because of globalization â€Å"international trade† takes place. International trade is the exchange of goods and services among different countries, no country is self sufficient and can’t produce all that it needs to survive, and hence the countries need to Export and import to meet their needs. With the help of m odern techniques, up to date procedures, contemporary practices, globalization and highly advanced transportation system, the International trade system is spreading really fast. in today’s world International trade is important for meeting the needs of the country, not every country can produce all that it wants so in order to meet their needs and demands the trade takes place. International trade can benefit the economy of the country by expanding the local market and increasing the variety of the goods and services available. International trade is the basic source of bringing â€Å"FOREX† in the country. Trade often increases competition and it helps in reducing monopolistic pricing and the cons that generate from that. It encourages local investors and manufactures to perform better and keep stable pricing in the market. International trade is one of the major sources of revenue for the country. By doing more exports and fewer imports the country can actually achi eve economic stability. international trade can help reduce local dependence on the existing companies and international trade can even help stabilize seasonal market fluctuations. No matter the level of the development of the country there will always be some specific products that other countries must be producing at a cheaper rate, in order to make maximum use of minimum resources the country import those certain goods, The are produced at lower marginal costs, this help countries save and stay in their budget, this concept is known as the â€Å"Comparative Advantage†. International trade is one of the best examples of Globalization. In spite of all these benefits international tra

Saudi Arabian Press Coverage of the Events in Bahrain Essay

Saudi Arabian Press Coverage of the Events in Bahrain - Essay Example I also wish to pay my sincere regards to Dr. Chris Paterson, who opened his door and mind for me. Moreover, I gratefully acknowledge the support of Mr. Turki Abdullah Al Sideri, as he stood with me throughout the MA degree program, and very special thanks go to Al Saud University for providing me with this wonderful opportunity. I am also extremely grateful to the staff members, friends, colleagues and all others who supported, taught and helped me in the research and in learning the language, both at Sheffield University and Leeds University. A great deal of love, loyalty, and thanks go to the government of my country for providing me with this amazing opportunity to complete the MA program. Last, but certainly not least, I would like to thank my father, mother, brothers, and sisters for their overwhelming support. This research paper investigates the performance of the Saudi Arabian press during the events in Bahrain between 14 February and 16 March 2011. It analyses the content of the press in Saudi Arabia with respect to the number of stories related to the Bahrain conflict and the theme and tone of these stories. Content analysis of the two leading Saudi Arabian newspapers, Alriyadh and Alyaum, was carried out by employing a constructed week approach for the entire period of the conflict. The results of this content analysis revealed that the coverage of the events in Bahrain by the Saudi Press was, to a great extent, dependent on the Saudi Press Agency (SPA), because of the constraints imposed on the press in the Kingdom by the Saudi Arabian information policy. The news related to Bahrain mostly appeared to be positive in tone and tended to support the agenda and interests of the government of Saudi Arabia with regards the political and democratic position of Bahrain, particularly when it involved the status of the Shiahs in the country.  

Brainscapes Case Study Example | Topics and Well Written Essays - 750 words

Brainscapes - Case Study Example Concentration of sodium (Na+) ions in the intracellular and extracellular space of tissues was the major identifier of hippocampus atrophy (with dead neurons). According to this case, brain tissues showing signs of Alzheimer’s disease (AD) had dead neurons and appear smaller than usual by expansion of the extracellular space and shrinkage of intracellular space. 2. Expected Sodium Concentration Changes The change of sodium concentration in the hippocampus region is used to differentiate healthy tissues from diseased ones in that unhealthy tissues have shrunk. It follows then that the hippocampus volume or processes are directly correlated with sodium concentration within and without the extracellular and intracellular space. Decreased volume of the hippocampus can possibly occur when neurons inside have died or are not functioning at their optimum levels. One expected change or difference is a difference in concentration of sodium ions in extracellular space and intracellular space commonly called a gradient difference. This differential concentration of sodium gradient normally occurs due to defects of Na+/K+ ATPase pump which may be blocked. Brain tissues afflicted by AD usually have a smaller volume than normal ones and identified by extracellular space that is expanded as well as shrinkage of the intracellular space. 3. Additional Information from Outside Sources Journal Article: Tissue Sodium Concentration in Human Brain Tumors as Measured with 23Na MR Imaging. Sourced from Radiology Journal Journal article: Brain tissue sodium concentration in multiple sclerosis: a sodium imaging study at 3 tesla. Sourced from National Center for Biotechnology Information database Journal Article: Na+ and K+ ion imbalances in Alzheimer's disease. Sourced from National Center for Biotechnology Information database Journal article: Distribution of Brain Sodium Accumulation Correlates with Disability in Multiple Sclerosis: A Cross-sectional 23Na MR Imaging Study. Sour ced from Radiology Journal 4. Outside Sources Contribution to the case The source (Ouwerkerk, Ronald, et al.) asserts that the concentration gradient difference of Na in tissues can be used to differentiate healthy tissues from affected one. Another source (Inglese, M., et al.) has findings that mechanisms of injury can be diagnosed through Na magnetic imaging. The third source (Vitvitsky V.M.) is of the opinion that failure of previous research come up with findings on cell homeostasis can be a basis of understanding development and progression of AD. Finally the last source (Zaaraoui, Wafaa, et al.) comes up with the finding that tissue injury can be diagnosed with NA imaging. 5. Case Study Finding that is Consistent with other Sources All the sources normally use differential concentration gradient of sodium in the intracellular space and extracellular space to come up with their findings. In this manner, atrophic tissues are identified from healthy ones after imaging. 6. Inconcl usive information with Sodium Concentration Changes Although using differential concentration to assay for atrophic tissues and organs can be very effective, it can also give wrong misdiagnosis especially when movement of sodium ions is impeded by other factors. For example, there can be sodium ions leakage due to a defective Na+K+ ATPase pump. 7. Resolution of the Case Sodium imaging can be a helpful technique

Qualitative Review on Psychological Intervention for Young People Research Paper

Qualitative Review on Psychological Intervention for Young People Living with HIV - Research Paper Example It aims to clarify and explicate the necessity of emotional and psychological intervention these young survivors of HIV to provide them meaningful life and motivation to undertake medical services and support to keep them living while bearing this health problem. It will illustrate how psychosocial intervention is provided to share hope for these children and teens, as well as, make them better persons despite everything. This is a qualitative review on psychosocial intervention for young people living with HIV, a transmissible disease that can infect a person through sexual contact or by other means known to many physicians. Researched materials generally used and employed survey, interviews and secondary materials in their studies to support and explicate the conditions of young populace living with HIV. This study is limited and focused only to young people living with HIV, thus, consider only the plight of children and teens. Kumar, Mmari, and Barnes (2012) pointed that there are already 1.7 billion young people within the age bracket of 10 to 24 years in the world that are infected with HIV disease and about 85% of them are living in developing countries. Kumar et al (2012) reported that the mortality rate of HIV infected persons are 3% and most of them reduced their chance of survival at the age of 60 in countries with high percentile of HIV-infected population. Nowadays, young people are the fastest-growing cohort of new HIV infections globally reaching about 40% of new HIV-infected people in 2007 (Kumar et al, 2012). There are about 5 million young people that are nowadays living with HIV and there are an estimated 5,000 youths aging 15 and 24 years that are infected everyday (Kumar et al, 2012). These global figures likely underestimate the total burden of HIV borne by young people, as there has been no systematic evaluation of the numbers of youth who are long-term survivors of perinatal infection. New evidence and estimations of HIV’s effect o n child mortality bared that about 13% of perinatally infected children can only survive up to the age of 10 years. But noting that the global interventions of prevention of mother-to-child transmission (PMTCT) programs has just been recently introduced in high-risk countries, it is always possible that cohort may contribute significantly to the increasing number of youths living with HIV. Albeit the rigorous efforts for HIV prevention, however the incessant sexuality and the social nature of all persons can increase the numbers of youth living with HIV. HIV-infected persons have dire needs for psychosocial support knowing that this can cause social stigma and can evoke innate anger or shock after being diagnosed positive thereof (AVERTing HIV & Aids, 2012). Relation with immediate families and friends will be altered, thus limits their social nature and level of interactions. Psychosocial therapy for HIV-infected persons can bring about positive outcome. Experts posit that this can help enhance a survivor response to health service; strengthen his behaviour while exercising preventive measures; and, mitigate the possibility that extreme depression may result into suicidal tendencies (AVERTing HIV & Aids, 2012). In a survey conducted by researchers in United States, psychosocial intervention was mentioned many times perceived as the most helpful measure that could help them live with HIV, especially as they progress their lives with

Heartsaver EAD Personal Statement Example | Topics and Well Written Essays - 750 words

Heartsaver EAD - Personal Statement Example My goals in taking this training were to learn basic lifesaving skills. These included CPR techniques for both adults and children, using barrier devices to conduct CPR, and how to stop someone from choking. I also wished to learn the signs of severe medical emergencies such as cardiac arrest, choking, heart attack, and stroke. I also wanted to learn about the Heartsaver AED, which is also known as an automated external defibrillator. I learned that it delivers a shock to the heart and can save the life of someone who is suffering from a heart attack, as well as learning how to use it. I wanted to be fully prepared to use the device in case I am ever in a situation where someone's life around me depends on it. Also, I wanted to polish my CPR skills. Learning these things was, to me, a way to show my community that I care about their safety and health and want to do my part to help protect it. It is difficult to pick a principle or two that stands out from the others, because they are all equally important to me. It does not matter if a person is having a heart attack or if they are choking: they still need someone fully trained to step in and save their life. I liked this course since I found that it taught me first aid on the four major life-threatening emergencies. References Heartsaver AED. (2008). American Heart Association. Retrieved February 18, 2008, from

Narrative Essay Example Essay Example for Free

Narrative Essay Example Essay Have you ever been in one of those never ending conversations? The ones where the speaker goes on and on for ages about a topic that you do not understand and could care even less about? Have you ever felt like a joke went straight over your head or that you were missing something as you struggled to find the context in a conversation? That is an everyday occurrence for people like me, affectionately called Aspies: people who have what is known as Asperger’s Syndrome. Since being diagnosed with this, everyone who has been aware of it has felt the need to make some sort of accommodation for my ‘disability. ’ A diagnosis that society feels I need because I think differently than the rest. How does society define you? I have spent my entire life trying to prove that our labels do not matter in comparison to our contributions to society. Aspies are very socially awkward. We cannot read non-verbal cues, societal niceties are often thought strange and hard to grasp, and we tend to be more than a little introverted. I have a very ‘mild’ case. No, I cannot read social cues. Yes, society’s unwritten rules drive me crazy. Absolutely, I would prefer to be alone or with a small select group of people. However, none of these characteristics define me. One characteristic of Aspies is that we often have a specialized and intense interest in something. My obsession is Star Trek, particularly the alternate reality movie series starring Chris Pine, Zachary Quinto, Zoe Saldana, Simon Pegg, and Karl Urban. I once heard someone equate people with Asperger’s to the Vulcan race from Star Trek. It is quite a fitting description. Vulcans are typically calm, rational, and even keeled people, but lord help you if you manage to anger one. They do not like to be touched and have a ‘muted’ sense of their own, and others, emotions (although in truth both are so sensitive that we have to shut off our empathy in order to function). Spock, the most commonly known Vulcan, exhibits this range of emotions in the JJ Abrams 2009 Star Trek remake movie. He is coolly rational, even as his planet is destroyed, but becomes near homicidal after Kirk starts throwing disparaging comments about his mother. Even so Captain Kirk and Mr. Spock are two-thirds of a trio that has gone down in pop culture legend along with the ever snarky Dr. McCoy. The Freudian trio that everyone so loves shows that there needs to be a balance of personalities which in the case of Star Trek, as in so many others, is the cold and logical (Spock), the emotional and humanistic (McCoy), and the rational and intuitive (Kirk). There are so many labels that get thrown on people throughout their lifetime; jarhead, slut, and geek just to name a few. Not many people strictly fit in to just the one singular box to which society relegates us. Really, who wants to fit in just one category? Every person is, as my friend Marilyn would say, a unique and beautiful snowflake. Having Asperger’s certainly qualifies me as a ‘special snowflake’, but there are some drawbacks. One of those is that we find it incredibly difficult to discuss our personal lives and often the only people who are aware of our personal thoughts and feelings are those who are in our close inner circle. One of my inner circle in high school was a girl named Jules. She was, without a doubt, the poster child for the school. She was beautiful, the head cheerleader who competed in beauty pageants and was the prom princess. It would have been so easy for her to have been content with being well loved by the community just because she was pretty and popular. Jules was not like that though. She was the vice president of our class three years running. She graduated a mere . 0002 from being the salutatorian. She was involved in the student community service club and the school’s religious advocacy team. Jules could have been content with her place as a cheerleader in the status quo, but she chose to defy society’s expectations of her. Within those societal labels is one of the most interesting phenomena; the labels are so generalized. Take for instance the geek or nerd box. It is a label that I accept as one of the closest fit for identifying me because I love to read, can quote passages of Harry Potter on a whim, and spend entirely too much of my time on FanFiction, just to name of few of my personal quirks. There are so many different ways that people are relegated to this outlier corner. Trekkies, Whovians, people who like anime and manga, movie nerds, and people who love working with technology are just small portion of the different kinds of people that are defined as a nerd. The labels that limit us so much do not even completely define us. They do not fully describe who we are as people or give full insight into our personalities. In my lifetime some of the most extroverted, party-hard people were nerds and some of the quietest and shy were cheerleaders. Bringing us full circle, I am an Aspie. However, I have gotten better with time and a little coaching at understanding social cues. I understand that the rant that I have been going on for the last three pages probably does not interest you. You have done the exact same things before. You have gone on and on about something that you are passionate about without regard for the interest level of those around you. I could go on for hours, yet sometimes being concise is better. I doubt there are many people who have not at least heard of The Breakfast Club. Its last remarks so poignantly drives the concept home: Brian Johnson: You see us as you want to see us In the simplest terms, in the most convenient definitions. But what we found out is that each one of us is a brain Andrew Clark: and an athlete Allison Reynolds: and a basket case Claire Standish: a princess John Bender: and a criminal Brian Johnson: Does that answer your question? Sincerely yours, the Breakfast Club. To you, who, whatever box you may have been stuck in or maybe even embraced all on your own, remember labels are just for cans of soup. The Breakfast Club. Dir. John Hughes. Perf. Emilio Estevez, Judd Nelson, Molly Ringwald. 1985. Universal, 2003. DVD.

The Nmc Code Of Conduct Nursing Essay

The Nmc Code Of Conduct Nursing Essay Nursing is a profession regulated by the Nursing and Midwifery Council (NMC 2008). The NMC is an organisation set up by the Parliament to protect the public by ensuring that nurses and midwives provide high standards of care to their patients. These healthcare professionals are also accountable for their own actions. The body sets standards for education, practice and conduct as well as providing advice for nurses and midwives. The NMC also considers allegations of misconduct or unfitness to practice. Using the case study given, it shall be the authors aim to demonstrate the understanding of the NMC Code of Conduct suggesting ways in which it can be applied to practice. In order to comply with the NMC Code of Conduct of confidentiality, the patient to be discussed in this assignment will be referred to as Mrs X. Furthermore the author will explore the four main principles of the code relating them to issues arising out of the case study. The author will also demonstrate the understan ding of ethical issues arising, analysing and discussing autonomy, non-maleficence, beneficence and justice. The case study refers to an 80 year old woman with a hip fracture, admitted to a hospital ward from a nursing home and urgently added to the operation list. She is bedridden, with severe heart problem and in early stages of Alzheimers disease and appeared to be coherent and lucid as recorded. She agreed to have a hip replacement operation after the consultant explained the procedure. On her way to theatre, she changes her mind and the consultant was informed. The consultant insists on proceeding, citing a busy week ahead and commenting that these elderly confused patients dont know their own mind. According to the NMC Code of Conduct, a healthcare professional has a duty to care and protect the interest of those in their care regardless of age, gender, culture, religious and political beliefs. Mrs X is 80 years old but the professionals still have a duty of care and must protect her interests. An interview was carried out by the medical staff and the patient appeared coherent and lucid but Mrs X has changed her mind on route to theatre. This author will critically examine the procedures that followed. The consultant explained the procedure to the patient who agreed to have the operation. Thompson et al (1994) stated that communication is one of the fundamental aspects in nursing The consultant was informed of the patients decision to change her mind on the operation and responds stating that We will have to proceed. As a nurse one could argue that the consultant should respect this decision as going against it would be breaching the NMC code. Mrs Xs decision to change her mind on the way to theatre, not wanting to go ahead with the operation should be respected. Hope et al (2008) stated that a patients autonomy can result in conflict, raise ethical dilemmas and may not be straight forward. Autonomy is defined as the right to choose or refuse treatment .Beauchamp and Childress (2009). The consultant could also argue that he is working in the best interest of the patient but does this override the patients right to make her own decision? Beauchamp and Childress (2009) stated that individuals views and rights must be respected as long as these individuals thoughts and action do not cause harm to other people. The NMC makes a point of highlighting the point of advocating for patients. In this instance, the nurse faces the ethical dilemma of standing up to the consultant and advocating for the patien t in order to uphold the code. Thompson (2003, cited in Buka, 2008) suggests that ethics is a study of how people behave, what they do, the reasons they give for their actions and the justification behind their decision. The need to maintain professional boundaries as well means that nurses have to raise their points in a manner that does not destabilise the team. Each and every member of the healthcare team must act as the patients advocate and remind or challenge colleagues should they fail to practice according to standards, Hindle and Coates (2011). If any medical team members working with the consultant on Mrs Xs case are not in agreement with his decision to proceed, they should challenge or remind him of the ethical code stipulating that the patients decision must be respected. When healthcare professionals are faced with dilemmas, patients should always come first. Childs et al (2009) states that when considering our actions we are bound by NMC codes, standards and guideline s, for students guidelines set by their training institution by local standards and guidelines within the clinical practice area and by the law of the country. It is unprofessional and unlawful to force treatment on anyone. Although the consultant explained the procedure, one could argue that making information available is different from effective communication. Consent was given the first time but the patient later changed her mind. An exploration for her reason to change her mind should have taken place and at least inform the patient that the operation was going ahead and the reasons for going ahead. The wording used by the consultant could be a concern. We will have to proceed. We have a very busy week ahead; these elderly confused patients dont know their own mind. Carry on as usual. One could interpret that the consultant is suggesting that when people get old, they automatically become confused; which could be stereotyping amounting to discrimination which is against the law. This could be taken to suggest that the consultant is of the assumption that the elderly are confused and dont know whats good for themselves and so should have decisions made for them. Patients are supposed to be treated with dignity, respect and as individuals considering their physical, psychological and social care with decisions made in partnership with clinicians, rather than by clinicians alone according to DOH (2010). Hendnrick (2004) defines consent as the permission given by patient voluntarily, without pressure, force or manipulation or undue influence. The NMC emphasises that healthcare professionals must seek consent from their patients otherwise they might be liable to be charged with assault or battery. The consent could either be in writing or verbal. In the event of a law suit, such documents and discussions can then be used in courts of law. Although Mrs X had given consent for the operation to proceed, health professionals should respect the withdrawal of consent. Proceeding with the planned operation against Mrs Xs wish amounts to violation of her rights and the nurse has a duty to highlight this aspect. The Mental Capacity Act (2005) was established to empower and to protect vulnerable people in making their own decisions. In particular, this was to safeguard those who lack capacity and those who have difficulties in making decisions because of illness, disability and those with mental health problems. The mental capacity act has four main principles of capacity: A person must be assumed to have capacity unless it is proved otherwise. Mrs X should be deemed to have capacity as she was interviewed and appeared coherent and lucid. Any act or decision taken on behalf of someone lacking capacity must be in the persons best interest. The consultant could argue he was working in the best interest of the patient. In the event that Mrs X lacks capacity, an advocate could be appointed to act on her behalf. An unwise decision is not to be taken as a lack of capacity. Even though Mrs X changed her mind and appears to have made an unwise decision, this should not be seen as lack of capacity. Until all practicable steps have been taken to help someone make a decision without success, they cannot be treated as lacking capacity. The consultant did not exhaust all efforts to help Mrs X in her decision making as no interaction took place after she changed her mind. The consultant took it upon himself to make the decision and dismissed Mrs X as an elderly confused patient who does not know her mind. The Mental Capacity Act (2005) has a test for capacity which states that a person lacks capacity if at the material time he is unable to make a decision for himself in relation to the matter because of an impairment of or a disturbance in the functioning of, the mind or brain. It does not matter whether the impairment or disturbance is temporal or permanent (Brammer, 2007). A person is unable to make a decision for himself if he is unable to understand the information relevant to the decision, to retain the information, to use or to weigh up that information as part of the process of making the decision, or to continue the decision (Brammer 2007). Section 3 of the act states that if the patient can retain information relevant to the decision for a short time only, this does not necessarily mean she cannot make a decision. When Mrs X changed her mind the consultant should have respected this decision because she was capable of retaining information for a while, had thought it through and decided she did not want to proceed. The Mental Health Act (1983) covers the reception, care and treatment of mentally disordered persons, the management of their property and other related matters. The act empowers authorities to detain those diagnosed with a mental disorder in hospital or police custody and have their disorder assessed or treated against their wishes, known as sectioning. Mrs X was diagnosed as having early signs of Alzheimers disease. This disease is a form of dementia, a neurologic disease characterized by loss of mental ability severe enough to interfere with normal activities of daily living. It usually occurs in old age, and is marked by a decline in cognitive functions such as remembering, reasoning, and planning. As Alzheimers disease is a progressive illness with no recovery, it is not applicable to use the Mental Health Act (1983) because whether or not treatment is given for the disease, this will not improve the decision making capacity of Mrs X. The General Medical Council clearly stipulates that healthcare professionals ought not to discriminate but should treat those in their care fairly based on their needs. The consultant is going against the GMCs code of conduct when he ignores the patients wish to discontinue with the operation. The GMC emphasises that patients have the right to change their minds on decisions.(ref) Nurses are required by the NMC Code of Conduct to express compassionate attitudes in their careers (Byrne and Byrne 1992). Nurses act as advocates for patients and as such can be described as special and unique to other health care professionals as they spend more time with the patients (Norman and Ryrie 2004). They are expected to develop a nurse-patient relationship which must be kept professional. It is also a nurses duty in accordance with NMC to educate the patient. Mrs X should have been educated and made aware of the advantages and disadvantages of the operation. The principle of non-maleficence is one which seeks to avoid intentional harm. Mrs X does not wish to undergo the procedure so to agree with her wish would be harmful although proceeding may harm any existing relationship between the healthcare professionals and the patient. What then happens if for instance the procedure does not go according to plan? Operations to correct hip fractures in the elderly are common and to abstain from conducting them would result in a lot of pain and discomfort not to mention the immobility issue. It is common knowledge that bedridden elderly patients if not moved regularly will develop pressure sores (Onslow 2005). The principle that requires action which benefits the patient is known as beneficence. To effect such an action sometimes medical professionals have to ignore the wishes of the patient if they can prove the patients incapacity to consent. While respecting the right of Mrs Xs treatment refusal, capacity test should be done to find out if she is capable of making her own decision. If Mrs X lacks capacity, then the medical staff should seek consent from the relatives or Independent advocates (Tingle and Cribb,. 2008). The ethical difficulties are compounded by such cases as the Canadian case of Malette v Schumann. The claimant came to hospital after being involved in a road accident. The doctor went ahead to perform blood transfusion despite the nurse having found a card in her pocket stating that she was a Jehovahs Witness and never to be given a blood transfusion. Later, on recovering the claimant won $20,000 of damages (Tingle and Cribb, 2008). The doctor was charged with battery. Mrs Xs wish not to proceed with the operation may be well founded and give grounds to litigation. The outcome of the operation also plays a major part in determining whether the decision to go ahead and operate is a good one or not. On tacking this assignment l learnt that establishing the patients consent is very vital for any action to be justifiable carried out. The consultant did manage to convince Mrs X to agree to undergo the procedure after talking her through it. He unfortunately could not accept her change of mind sighting her age as the problem. I felt that Mrs X hadnt been given enough time to ponder the idea of undergoing the procedure. She has been admitted to the hospital ward and urgently added to the list. I thought because she was in pain, she was not thinking straight and was pressured into giving consent. Looking back l now feel the consultant wanted the hip fracture operation to proceed as soon as possible as this would in turn ensure speed recovery. Looking at her age, I would like to think that the sooner she got operated on the quicker the recovery. He had the patients interest at heart. At the time l felt team work and better communication would have brought about better decision. The team members should have objected or aired their feelings against the consultants wish to proceed without consent. The positive was that if Mrs X was operated on, the pain would easy and she would then be mobile, which would be good for her heart. Taking the age issue into perspective the sooner she underwent the procedure the sooner she was expected to heal. The negative was that if anything went wrong, bearing in mind Mrs X had severe heart problem, the whole team would be in trouble. When Mrs X changed her mind about undergoing the procedure the issue should have been addressed properly since consent is fundamental in a patients care. A meeting between the medical care professionals to look into the reason of change of plan, if need be, a mental capacity test taken as is warranted under the Medical Health Capacity Act. In nursing the interests of the patients always come first. I think communication is vital in nursing. Communication is very important when dealing with patients in nursing. The consultant did not act as a professional when Mrs X changed her mind that she is not ready for the hip operation. I was not comfortable with his response as it sounded harsh, commanding and unprofessional when he was informed of Mrs X decision I have learnt that team is important in nursing and healthcare professionals should always respect the rights of their patients and consent is at the centre of every action. The author has explored the professional, legal and ethical implications of the case study provided. It has been identified that although the NMC provides guidance and regulates the nursing profession, the onus is on the practitioner to make decisions based on the guidelines. Although the nurses and doctors may be working together, it has also been noted the two professions are governed by two different bodies and therefore have different codes of ethics although some of the codes could be similar. The NMC code of conduct is often updated as the code sometimes conflicts with other policies and procedures from employment and the law. Nurses should ensure they are up to date with any changes and guidelines within this body (Beech 2007). Because of the trust accorded nurses by society (gained through recognition of nurses expertise) and the right given the profession to regulate practice (professional autonomy) individual clinicians and the profession must be both responsible and accoun table Hitchcock et al (2003). The basic ethical principles of beneficence, nonmaleficence, justice and autonomy which are among the ethical principles that influence decisions in health care ethics have been explored and applied to the case study. The Mental Capacity Act (2005) has also been discussed and identified as the main legal instrument relating to this case study. It is crucial that nurses understand how the law influences nursing practice, particularly in relation to anticipating lack of capacity Hindle and Coates (2011).