Number Theory for Mathematical Contests David A. SANTOS [email protected] August 13, 2005 REVISION Contents Preface iii 5 Linear Diophantine Equations 48 5.1 Euclidean Algorithm ............. 48 1 Preliminaries 1 Practice ....................... 50 1.1 Introduction .................. 1 5.2 Linear Congruences .............. 51 1.2 Well-Ordering ................. 1 Practice ....................... 52 5.3 A theorem of Frobenius ............ 52 Practice ....................... 3 Practice ....................... 54 1.3 Mathematical Induction ............ 3 5.4 Chinese Remainder Theorem ......... 55 Practice ....................... 7 Practice ....................... 56 1.4 Fibonacci Numbers .............. 9 Practice ....................... 11 6 Number-Theoretic Functions 57 6.1 Greatest Integer Function ........... 57 1.5 Pigeonhole Principle ............. 13 Practice ....................... 60 Practice ....................... 14 6.2 De Polignac’s Formula ............ 62 Practice ....................... 64 2 Divisibility 17 6.3 Complementary Sequences .......... 64 2.1 Divisibility .................. 17 Practice ....................... 65 Practice ....................... 18 6.4 Arithmetic Functions ............. 66 2.2 Division Algorithm .............. 19 Practice ....................... 68 Practice ....................... 20 6.5 Euler’s Function. Reduced Residues ..... 69 2.3 Some Algebraic Identities ........... 21 Practice ....................... 72 6.6 Multiplication in Z .............. 73 Practice ....................... 23 n Practice ....................... 75 6.7 Möbius Function ............... 75 3 Congruences. Zn 26 Practice ....................... 76 3.1 Congruences ................. 26 Practice ....................... 30 7 More on Congruences 78 3.2 Divisibility Tests ............... 31 7.1 Theorems of Fermat and Wilson ....... 78 Practice ....................... 32 Practice ....................... 80 7.2 Euler’s Theorem ................ 81 3.3 Complete Residues .............. 33 Practice ....................... 83 Practice ....................... 33 8 Scales of Notation 84 4 Unique Factorisation 34 8.1 The Decimal Scale .............. 84 4.1 GCD and LCM ................ 34 Practice ....................... 86 Practice ....................... 38 8.2 Non-decimal Scales .............. 87 4.2 Primes ..................... 39 Practice ....................... 88 8.3 A theorem of Kummer ............ 89 Practice ....................... 41 4.3 Fundamental Theorem of Arithmetic ..... 41 9 Miscellaneous Problems 91 Practice ....................... 45 Practice ....................... 93 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. The purpose of the course was to familiarise the pupils with contest-type problem solving. Thus the majority of the prob- lems 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. I would like to thank Eric Friedman for helping me with the typing, and Carlos Murillo for proofreading the notes. Due to time constraints, these notes are rather sketchy. Most of the motivation was done in the classroom, in the notes I presented a rather terse account of the solutions. I hope some day to be able to 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 Theory that is useful in contests. I also wrote notes (which I have not transcribed) dealing with primitive roots, quadratic reciprocity, diophantine equations, and the geometry of numbers. I shall finish writing 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 [email protected] iii Legal Notice This material may be distributed only subject to the terms and conditions set forth in the Open Publication License, version 1.0 or later (the latest version is presently available at http://www.opencontent.org/openpub/ . THIS WORK IS LICENSED AND PROVIDED “AS IS” WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IM- PLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE OR A WARRANTY OF NON-INFRINGEMENT. THIS DOCUMENT MAY NOT BE SOLD FOR PROFIT OR INCORPORATED INTO COMMERCIAL DOCUMENTS WITHOUT EXPRESS PERMISSION FROM THE AUTHOR(S). THIS DOCUMENT MAY BE FREELY DISTRIBUTED PROVIDED THE NAME OF THE ORIGINAL AUTHOR(S) IS(ARE) KEPT AND ANY CHANGES TO IT NOTED. iv Chapter 1 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 fascinated Man, who has been drawn to them either for their utility at solving practical problems (like those of measuring, counting sheep, etc.) or as a fountain of solace. Number Theory is one of the oldest and most beautiful branches of Mathematics. It abounds in problems that yet simple to state, 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. Are there infinitely many primes that are 1 more than the square 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 Well-Ordering The set N = 0,1,2,3,4,... of natural numbers is endowed with two operations, addition and multiplication, that satisfy the following properties{ 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-Ordering Axiom) Every non-empty subset S of the natural numbers has a least element. As an example of the use of the Well-Ordering Axiom, let us prove that there is no integer between 0 and 1. 2 Example Prove that there is no integer in the interval ]0;1[. 1 2 Chapter 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 m2 which 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 A rational number is a number which can be expressed as the ratio of two integers a,b, where b = 0. We denote the set of b 6 rational 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. 3 Example Prove that √2 is irrational. a Solution: The proof is by contradiction. Suppose that √2 were rational, i.e., that √2 = for some integers a,b. This implies b that the 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 equation and with max(a,b,c) > 0 6 6 6 6 6 6 as small as possible. If a + 2b = 4c then a must be even, a = 2a1. This leads to 32a1 + b = 2c . Hence b = 2b1 and so 6 6 6 6 6 6 16a1 + 32b1 = c . This gives c = 2c1, and so a1 + 2b1 = 4c1. But clearly max(a1,b1,c1) < max(a,b,c). This means that all of these must be zero. a2 + b2 a2 + b2 5 Example (IMO 1988) If a,b are positive integers such that is an integer, then is a perfect square. 1 + ab 1 + ab a2 + b2 Solution: Suppose that = k is a counterexample of an integer which is not a perfect square, with max(a,b) as small as 1 + ab possible. 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. 2 2 2 Now, a + b − k(ab + 1) = 0 is a quadratic in b with sum of the roots ka and product of the roots a − k.