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Challenge your problemsolving aptitude in number theory with powerful problems that have concrete examples which reflect the potential and impact of theoretical results. Each chapter focuses on a fundamental concept or result, reinforced by each of the subsections, with scores of challenging problems that allow you to comprehend number theory like never before. All students and coaches wishing to excel in math competitions will benefit from this book as will mathematicians and adults who enjoy interesting mathematics.
NUMBER THEORY: CONCEPTS AND PROBLEMS NUMBER THEORY: CONCEPTS AND PROBLEMS Titu Andreescu Gabriel Dospinescu Oleg Mushkarov Library of Congress Control Number: 2017940046 ISBN-10: 0-9885622-0—0 ISBN-13: 978-0-9885622—0—2 © 2017 XYZ Press, LLC All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (XYZ Press, LLC, 3425 Neiman Rd., Plano, TX 75025, USA) and the authors except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of tradenames, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights www.awesomemath.org Cover design by Iury Ulzutuev FORWARD PREDA MIHAILESCU Exercises are in mathematics like a vitalizer: they strengthen and train the elasticity of the mind, teach a variety of successful methods for approaching specific problems, and enrich the professional culture with interesting questions and results For a good treatment of a theory, examples and exercises are the art of presenting concrete applications, reflecting the strength and potential of the theoretical results A strong theory explained only by simple exercise often may reduce the motivation of the reader At the other end, there is a wide reserve of problems and exercises of elementary looking nature, but requiring vivid mind and familiarity with a good bag of tricks, problems of styles which were much developed by the interest that mathematical competition attracted worldwide in the last 50 years These problems can only loosely be ordered into applications of individual theories of mathematics, their flavor and interest relaying in the way they combine different areas of knowledge with astute techniques of solving Often, not always, the problems addressed have some deeper interest of their own and can very well be encountered as intermediate steps in the development of mathematical theories From this perspective, a good culture of problems can be to a mathematician as helpful, as the familiarity with classical situations in chess matches, to a professional chess player: they develop the aptitude to recognize, formulate and solve individual problems that may play a crucial role in theories and proofs of deeper significance The book at hand is a powerful collection of competition problems with number theoretical flavor They are generally grouped according to common aspects, related to topics like Diaisibility, GOD and LCM, decomposition of polynomials, Congruences and p-adz'c valuations, etc And these aspects can be found in the problems discussed in the respective chapter — beware though to expect much connection to the typical questions one would find in an introductory textbook to number theory, at the chapters with the same name The problems here are innovative findings and questions, and the connection is more often given by the methods used for the solution, than by the very nature of the problem ii Forward Some problems have a simple combinatorial charm of their own, without requiring much more than good observation — for instance (p 512, N 25), Find all m, n,p e Q>o such that all of the numbers m + i, n + i, p + Fl; are integers Others appear even weird at a first glance, like (p 656, N 8): For coprime positive integers p, q, prove that: -1 E(_1)lk/pJ+Lk/q1= 19:0 If m ls even , if pq is odd ’ or (N 36, p 543), requiring to show that infinitely many primes are coprime to the terms of the polynomially recursive sequence given by a1 = and an.” = (a3, + 1)2 — a% When one then does the homework, one notices that several useful and non trivial notions about floors are required for solving the problem The book also contains some basic propositions, which are in big part classical theorems, but also more specialized results, that can be applied for solving further problems Thus, beyond the spontaneous charm of some of the exercises, most problems are involved and require a good combination of solid understanding of the theoretical basics, with a good experience in problem solving Working through the book one learns a lot Do you want to know more on how large the difference between the product of k consecutive integers and their LCM can become? A series of results will provide an answer — and you will then certainly find also a set of variations of this theme For primes p, the Fermat quotient ¢(2) = ”+34 mod p has a well known development in terms of harmonic sums But if you want to know higher terms in its p-adic development, you can find them in the chapter on p-adic values Together with a series of less known, classical congruences of higher order of Wolfenstone, Morley, Ljunggren et al., this leads to a series of interesting questions and problems Not all problems are atomic training subjects; at the contrary, by a good choice of the problems, the authors may group elementary results that lead to remarkable understanding of some flmdamental number theoretical functions, like 71', a, 7', ¢ — the prime distribution flmction, the number of divisors and their sum, and the Euler totient, respectively Here also, if you want for instance to Forward iii understand how it happens that the fibers of the inverse ¢_1(X) of the Euler totient may become indefinitely large, several exercises lead to the understanding of this phenomenon It will not surprise that among the authors or solvers of the problems presented, one encounters numerous famous mathematicians, from classical to contemporaneous, ranging from Gauss, Lagrange, Euler and Legendre, through V Lebesgue, Lucas, but also Hurwitz and, unsurprisingly, Erdc’is and Schinzel: the borders between research mathematics and advanced problem solving are fluid This very short and selective overview of the book should have already suggested that the book can be read with various attitudes and expectations, and there is always much to profit from it The reader may traverse entire chapters of the book and get familiar with the specifics of the posed problems, but should definitely invest the time for trying to solve at least two or three problems alone, each time when working again with this book In spite of the well structured construction of the book, one can easily jump to chapters or sections of interest — they are to a large extent self-consistent And if not, good references help to find the necessary facts which were discussed at previous places of the book Altogether — while students eager to acquire experience helping to reach outstanding performance in mathematical competitions will profit most from this book, it is certainly a good companion both for professional mathematicians and for any adult with an active interest in mathematics Each one of them will find it a leisure to read and work over and over again through the problems of this book Preda Mihailescu Gottingen, May 2017 Mathematisches Institut der Universitat Gottingen E—mail: preda©uni—math.gwdg.de Contents Forward i Introduction Divisibility 2.1 Basic properties 2.1.1 Divisibility and congruences 2.1.2 Divisibility and order relation 2.2 Induction and binomial coefficients 2.2.1 Proving divisibility by induction 2.2.2 Arithmetic of binomial coefficients 2.2.3 Derivatives and finite differences 2.2.4 The binomial formula 2.3 Euclidean division 2.3.1 The Euclidean division 2.3.2 Combinatorial arguments and complete residue systems 2.4 Problems for practice 3 10 22 22 26 34 38 43 43 47 56 GOD and LCM 3.1 Bézout’s theorem and Gauss’ lemma 3.1.1 Bézout’s theorem and the Euclidean algorithm 3.1.2 Relatively prime numbers 3.1.3 Inverse modulo n and Gauss’ lemma 3.2 Applications to diophantine equations and approximations 3.2.1 Linear diophantine equations 63 63 63 68 72 80 80 vi Contents 3.3 3.4 3.2.2 Pythagorean triples 83 3.2.3 The rational root theorem 92 3.2.4 Farey fractions and Pell’s equation 96 Least common multiple 113 Problems for practice 121 The fundamental theorem of arithmetic Composite numbers 4.2 The fundamental theorem of arithmetic 134 4.2.1 The theorem and its first consequences 134 4.2.2 4.2.3 4.3 129 4.1 129 The smallest and largest prime divisor 144 Combinatorial number theory 149 Infinitude of primes 4.3.1 Looking for primes in classical sequences 4.3.2 Euclid’s argument 4.3.3 Euler’s and Bonse’s inequalities 154 155 160 171 4.4 Arithmetic functions 178 4.5 4.4.1 Classical arithmetic functions 4.4.2 Multiplicative functions 4.4.3 Euler’s phi function 4.4.4 The Mobius function and its applications 4.4.5 Application to squarefree numbers Problems for practice 178 184 194 206 210 216 Congruences involving prime numbers 225 5.1 Fermat’s little theorem 225 5.2 5.3 5.1.1 Fermat’s little theorem and (pseudo-)primality 225 5.1.2 Some concrete examples 230 5.1.3 Application to primes of the form 4k + and 3k + 238 Wilson’s theorem 5.2.1 Wilson’s theorem as criterion of primality 5.2.2 Application to sums of two squares Lagrange’s theorem and applications 5.3.1 The number of solutions of polynomial congruences 5.3.2 The congruence a:”’5 (mod p) 244 244 252 259 259 266 672 Chapter Solutions to practice problems Conversely, assume that p is a Fermat prime, then 90(1) — 1) = %1 since p — is a power of Since any primitive root mod p is a quadratic nonresidue mod p and since there are as many quadratic non-residues mod p as primitive roots mod p, it follows that every quadratic non-residue mod p is a primitive root mod p The result follows III 34 Let Mn) be the least positive integer k: such that wk E (mod n) for all a: relatively prime to n Prove that a) If k is a positive integer such that 13’“ E (mod n) for all in relatively prime to n, then k is a multiple of Mn) b) a) = lcm(Mm), Mn)) for m, n relatively prime 0) We have Mn) = Then for all a: relatively prime to n we have E 73" = (x’\(”))q - at" E 56’" (mod n) This contradicts the minimality of Mn) b) Let M = lcm(Mm), Mn)) Suppose that a: is relatively prime to mn, so it is relatively prime to both m and n Now by definition of M we have xM E (mod n) and xM E (mod m), thus wM E (mod mn), since gcd(m, n) = Since a: was arbitrary, this yields (thanks to a)) a) | M To prove that M | a), it suffices, thanks to a) and to symmetry in m and n, to prove that 9:a) E (mod n) for all :1: relatively prime to n Take such 11: Note that :1: is not necessarily prime to m, but since gcd(m,n) = we can find 3] such that y E a; (mod n) and y E (mod m) (Chinese remainder theorem) Now y is relatively prime to mn, so m”) E (mod n) But clearly m") E :r’\(m"), hence the result 8.6 Congmences for composite moduli 673 0) Note that in all cases Mn) | f (a+x) is a solution for all integers a, we deduce that all solutions of the problem (when n + is prime) are of the form f (m) = agw (mod n + 1) for some a E {1, 2, , n} and some primitive root modulo n + Conversely, it is easy to see that these are indeed solutions of the problem [I Bibliography [1] V Boju, L Funar, The Math Problems Notebook, Birkhauser, 2007 [2] Z I Borevich, I R Shafarevich, Number Theory, Academic Press (New York), 1966 [3] H Davenport, Multiplicative Number Theory, 2nd ed., Springer-Verlag (New York), 1980 [4] T Andreescu, G Dospinescu, Problems from the Book, XYZ Press, 2008 [5] C F Gauss, Disquisitiones Arithmeticae {Discourses on Arithmetic), English ed., Yale University Press (New Haven), 1966 [6] G H Hardy, E M Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press (Oxford), 1979 [7] K Ireland, M Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag (New York), 1982 [8] E Landau, Elementary Number Theory, Chelsea (New York), 1958 [9] T Nagell, Introduction to Number Theory, Chelsea (New York), 1981 [10] I Niven, H S Zuckerman, H L Montgomery, An Introduction to the Theory of Numbers, fifth edition, John Wiley Sons, Inc [11] G Pélya, G Szego, Problems and Theorems in Analysis, Vol I, SpringerVerlag (New York), 1972 684 Bibliography [12] G Pélya, G Szegé, Problems and Theorems in Analysis, Vol II, SpringerVerlag (New York), 1976 [13] K H Rosen, Elementary Number Theory and Its Applications, Addison— Wesley (Reading), 1984 [14] W Sierpinski, Elementary Theory of Numbers, Polski Academic Nauk, Warsaw, 1964 [15] W Sierpinski, 250 Problems in Elementary Number Theory, American Elsevier Publishing Company, Inc., New York, Warsaw, 1970 [16] J P Serre, A Course In Arithmetic, Springer-Verlag (New York), 1973 Other Books from XYZ Press 685 Other Books from XYZ Press Andreescu, T., Elliott, S., 114 Exponent and Logarithm Problems from the AwesomeMath Summer Program, 2017 Andreescu, T., Cri§an, V., Mathematical Induction A powerful and elegant method of proof, 2017 Andreescu, A., Andreescu, T., Mushkarov, 0., 113 Geometric Inequali- ties from the AwesomeMath Summer Program, 2017 Bosch, R., Cuban Mathematical Olympiads (2001-2016), 2017 Matei, V., Reiland, E., 112 Combinatorial Problems from the AwesomeMath Summer Program, 2016 Andreescu, T., Mortici, C., Tetiva, M., Pristine Landscapes in Elementary Mathematics, 2016 Andreescu, A., Vale, V., 111 Problems in Algebra and Number Theory, 2016 Andreescu, T., Mathematical Reflections - two special years, 2016 Andreescu, T., Pohoata, C., Korsky, S., Lemmas in Olympiad Geometry, 2016 10 Mihalescu, C., The Geometry of Remarkable Elements Points, lines, and circles, 2016 11 Andreescu, T., Pohoata, C., 110 Geometry Problems for the International Mathematical Olympiad, 2015 12 Andreescu, T., Boreico, I., Mushkarov, 0., Nikolov, N., Topics in Func— tional Equations, 2nd edition, 2015 13 Andreescu, T., Ganesh, A., 10.9 Inequalities from the AwesomeMath Summer Program, 2015 686 Other Books from X YZ Press 14 Andreescu, T., Pohoata, C., Mathematical Reflections - two great years, 2014 15 Andreescu, T., Ganesh, A., 108 Algebra Problems from the AwesomeMath Year-Round Program, 2014 16 Andreescu, T., Kisaéanin, B., Math Leads for Mathletes — a rich resource for young math enthusiasts, parents, teachers, and mentors, Book 1, 2014 17 Becheanu, M., Enescu, B., Balkan Mathematical Olympiads — the first 30 years, 2014 18 Andreescu, T., Mathematical Reflections — two more years, 2013 19 Andreescu, T., Rolinek, M., Tkadlec, J , 107 Geometry Problems from the AwesomeMath Year-Round Program, 2013 20 Andreescu, T., Rolinek, M., Tkadlec, J , 106 Geometry Problems from the AwesomeMath Summer Program, 2013 21 Andreescu, T., 105 Algebra Problems from the AwesomeMath Summer Program, 2013 22 Andreescu, T., Kane, J., Purple Comet Math Meet! - the first ten years, 2013 23 Andreescu, T., Mathematical Reflections — the next two years, 2012 24 Andreescu, T., Dospinescu, G., Straight from the Book, 2012 25 Andreescu, T., Mathematical Reflections — the first two years, 2011 26 Andreescu, T., Dospinescu, G., Problems from the Book, 2nd edition, 2010