Elementary Number Theory Notes c (cid:13) David A. Santos January 15, 2004 ii Contents Preface v 1 Preliminaries 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Well-Ordering . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Mathematical Induction . . . . . . . . . . . . . . . . . . . 4 1.4 Binomial Coefficients . . . . . . . . . . . . . . . . . . . . . 16 1.5 Vie`te’s Formulæ . . . . . . . . . . . . . . . . . . . . . . . 16 1.6 Fibonacci Numbers . . . . . . . . . . . . . . . . . . . . . 16 1.7 Pigeonhole Principle . . . . . . . . . . . . . . . . . . . . . 23 2 Divisibility 31 2.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 Some Algebraic Identities . . . . . . . . . . . . . . . . . . 38 3 Congruences. Z 47 n 3.1 Congruences . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Divisibility Tests . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3 Complete Residues . . . . . . . . . . . . . . . . . . . . . . 60 4 Unique Factorisation 63 4.1 GCD and LCM . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Fundamental Theorem of Arithmetic . . . . . . . . . . . 76 iii iv CONTENTS 5 Linear Diophantine Equations 89 5.1 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . 89 5.2 Linear Congruences . . . . . . . . . . . . . . . . . . . . . 94 5.3 A theorem of Frobenius . . . . . . . . . . . . . . . . . . . 96 5.4 Chinese Remainder Theorem . . . . . . . . . . . . . . . . 100 6 Number-Theoretic Functions 105 6.1 Greatest Integer Function . . . . . . . . . . . . . . . . . . 105 6.2 De Polignac’s Formula . . . . . . . . . . . . . . . . . . . . 116 6.3 Complementary Sequences . . . . . . . . . . . . . . . . 119 6.4 Arithmetic Functions . . . . . . . . . . . . . . . . . . . . . 121 6.5 Euler’s Function. Reduced Residues . . . . . . . . . . . . 128 6.6 Multiplication in Z . . . . . . . . . . . . . . . . . . . . . . 134 n 6.7 Mo¨bius Function . . . . . . . . . . . . . . . . . . . . . . . 138 7 More on Congruences 141 7.1 Theorems of Fermat and Wilson . . . . . . . . . . . . . . 141 7.2 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 147 8 Scales of Notation 151 8.1 The Decimal Scale . . . . . . . . . . . . . . . . . . . . . . 151 8.2 Non-decimal Scales . . . . . . . . . . . . . . . . . . . . . 157 8.3 A theorem of Kummer . . . . . . . . . . . . . . . . . . . . 161 9 Diophantine Equations 165 9.1 Miscellaneous Diophantine equations . . . . . . . . . . 165 10 Miscellaneous Examples and Problems 169 10.1 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . 170 11 Polynomial Congruences 173 12 Quadratic Reciprocity 175 13 Continued Fractions 177 Preface These notes started in the summer of 1993 when I was teaching Number Theory at the Center for Talented Youth Summer Program at the Johns Hopkins University. The pupils were between 13 and 16 years of age. Thepurposeofthecoursewastofamiliarisethepupilswithcontest- type problem solving. Thus the majority of the problems are taken from well-known competitions: AHSME American High School Mathematics Examination AIME American Invitational Mathematics Examination USAMO United States Mathematical Olympiad IMO International Mathematical Olympiad ITT International Tournament of Towns MMPC Michigan Mathematics Prize Competition (UM)2 University of Michigan Mathematics Competition STANFORD Stanford Mathematics Competition MANDELBROT Mandelbrot Competition Firstly, I would like to thank the pioneers in that course: Samuel Chong, Nikhil Garg, Matthew Harris, Ryan Hoegg, Masha Sapper, Andrew Trister, Nathaniel Wise and Andrew Wong. I would also like to thank the victims of the summer 1994: Karen Acquista, Howard Bernstein, Geoffrey Cook, Hobart Lee, Nathan Lutchansky, David Ripley, Eduardo Rozo, and Victor Yang. IwouldliketothankEricFriedmanforhelpingmewiththetyping, and Carlos Murillo for proofreading the notes. Due to time constraints, these notes are rather sketchy. Most of v vi CONTENTS themotivationwasdoneintheclassroom,inthenotesIpresenteda ratherterseaccountofthesolutions. Ihopesomedaytobeableto give more coherence to these notes. No theme requires the knowledge of Calculus here, but some of the solutions given use it here and there. The reader not knowing Calculus can skip these problems. Since the material is geared to High School students (talented ones, though) I assume very little mathematical knowledge beyond Algebra and Trigonometry. Here and there some of the problems might use certain properties of the complex numbers. A note on the topic selection. I tried to cover most Number The- ory that is useful in contests. I also wrote notes (which I have not transcribed) dealing with primitive roots, quadratic reciprocity, dio- phantineequations, andthegeometryofnumbers. Ishallfinishwrit- ing them when laziness leaves my weary soul. I would be very glad to hear any comments, and please forward me any corrections or remarks on the material herein. David A. Santos 1 Chapter Preliminaries 1.1 Introduction We can say that no history of mankind would ever be complete without a history of Mathematics. For ages numbers have fasci- nated Man, who has been drawn to them either for their utility at solvingpracticalproblems(likethoseofmeasuring,countingsheep, etc.) or as a fountain of solace. Number Theory is one of the oldest and most beautiful branches ofMathematics. Itaboundsinproblemsthat yet simpletostate, are very hard to solve. Some number-theoretic problems that are yet unsolved are: 1. (Goldbach’s Conjecture) Is every even integer greater than 2 the sum of distinct primes? 2. (Twin Prime Problem) Are there infinitely many primes p such that p+2 is also a prime? 3. Arethereinfinitelymanyprimesthatare1morethanthesquare of an integer? 4. Is there always a prime between two consecutive squares of integers? In this chapter we cover some preliminary tools we need before embarking into the core of Number Theory. 1 2 Chapter 1 1.2 Well-Ordering The set N = {0,1,2,3,4,...} of natural numbers is endowed with two operations,additionandmultiplication,thatsatisfythefollowingprop- erties for natural numbers a,b, and c: 1. Closure: a+b and ab are also natural numbers. 2. Associative laws: (a+b)+c = a+(b+c) and a(bc) = (ab)c. 3. Distributive law: a(b+c) = ab+ac. 4. Additive Identity: 0+a = a+0 = a 5. Multiplicative Identity: 1a = a1 = a. One further property of the natural numbers is the following. 1 Axiom Well-OrderingAxiomEverynon-emptysubsetS ofthenat- ural numbers has a least element. AsanexampleoftheuseoftheWell-OrderingAxiom,letusprove that there is no integer between 0 and 1. 2 Example Prove that there is no integer in the interval ]0;1[. Solution: Assume to the contrary that the set S of integers in ]0;1[ is non-empty. Being a set of positive integers, it must contain a least element, say m. Now, 0 < m2 < m < 1, and so m2 S. But this is ∈ saying that S has a positive integer m2which is smaller than its least positive integer m. This is a contradiction and so S = ∅. We denote the set of all integers by Z, i.e., Z = {...−3,−2,−1,0,1,2,3,...}. A rational number is a number which can be expressed as the ratio a of two integers a,b, where b = 0. We denote the set of rational b 6 numbers by Q. An irrational number is a number which cannot be expressed as the ratio of two integers. Let us give an example of an irrational number. Well-Ordering 3 3 Example Prove that √2 is irrational. Solution: The proof is by contradiction. Suppose that √2 were ra- a tional, i.e., that √2 = for some integers a,b. This implies that the b set A = {n√2 : both n and n√2 positive integers} is nonempty since it contains a. By Well-Ordering A has a smallest element, say j = k√2. As √2−1 > 0, j(√2−1) = j√2−k√2 = (j−k)√2 is a positive integer. Since 2 < 2√2 implies 2 − √2 < √2 and also j√2 = 2k, we see that (j−k)√2 = k(2−√2) < k(√2) = j. Thus (j − k)√2 is a positive integer in A which is smaller than j. This contradicts the choice of j as the smallest integer in A and hence, finishes the proof. 4 Example Let a,b,c be integers such that a6+2b6 = 4c6. Show that a = b = c = 0. Solution: Clearly we can restrict ourselves to nonnegative numbers. Choose a triplet of nonnegative integers a,b,c satisfying this equa- tion and with max(a,b,c) > 0 as small as possible. If a6+ 2b6 = 4c6 then a must be even, a = 2a . 1 This leads to 32a6+b6 = 2c6. Hence b = 2b and so 16a6+32b6 = c6. 1 1 1 1 This gives c = 2c , and so a6+2b6 = 4c6. But clearly max(a ,b ,c ) < 1 1 1 1 1 1 1 max(a,b,c). This means that all of these must be zero. a2+b2 5 Example (IMO 1988) If a,b are positive integers such that is 1+ab a2+b2 an integer, then is a perfect square. 1+ab 4 Chapter 1 a2+b2 Solution: Supposethat = kisacounterexampleofaninteger 1+ab whichisnotaperfectsquare,withmax(a,b)assmallaspossible. We may assume without loss of generality that a < b for if a = b then 2a2 0 < k = < 2, a2+1 which forces k = 1, a perfect square. Now,a2+b2−k(ab+1) = 0isaquadraticinbwithsumoftheroots kaandproductoftherootsa2−k.Letb ,bbeitsroots, sob +b = ka 1 1 and b b = a2−k. 1 Asa,karepositiveintegers,supposingb < 0isincompatiblewith 1 a2+b2 = k(ab +1). As k is not a perfect square, supposing b = 0 is 1 1 1 incompatible with a2+02 = k(0 a+1). Also · a2−k b2−k b = < < b. 1 b b a2+b2 Thuswehavefoundanotherpositiveintegerb forwhich 1 = k 1 1+ab 1 and which is smaller than the smallest max(a,b). This is a contradiction. It must be the case, then, that k is a perfect square. Ad Pleniorem Scientiam 6 APS Find all integer solutions of a3+2b3 = 4c3. 7 APS Prove that the equality x2+y2+z2 = 2xyz can hold for whole numbers x,y,z only when x = y = z = 0. 1.3 Mathematical Induction The Principle of Mathematical Induction is based on the following fairly intuitive observation. Suppose that we are to perform a task that involves a certain number of steps. Suppose that these steps must be followed in strict numerical order. Finally, suppose that we knowhowtoperformthen-thtaskprovidedwehaveaccomplished

Elementary Number Theory Notes

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