i ”God made the integers, all else is the work of man.” Leopold Kronecker ii NUMBER THEORY Structures, Examples, and Problems Titu Andreescu Dorin Andrica ii Contents Foreword 7 Acknowledgments 9 Notation 11 I STRUCTURES, EXAMPLES, AND PROBLEMS 13 1 Divisibility 15 1.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 The greatest common divisor . . . . . . . . . . . . . . . . . 30 1.4 Odd and even . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.5 Modular arithmetics . . . . . . . . . . . . . . . . . . . . . . 42 1.6 Chinese remainder theorem . . . . . . . . . . . . . . . . . . 47 1.7 Numerical systems . . . . . . . . . . . . . . . . . . . . . . . 50 1.7.1 Representation of integers in an arbitrary base . . . 50 1.7.2 Divisibility criteria in the decimal system . . . . . . 51 2 Contents 2 Powers of Integers 61 2.1 Perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.2 Perfect cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 70 2.3 k th powers of integers, k ≥ 4 . . . . . . . . . . . . . . . . . . 72 3 Floor Function and Fractional Part 77 3.1 General problems . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Floor function and integer points . . . . . . . . . . . . . . . 83 3.3 An useful result . . . . . . . . . . . . . . . . . . . . . . . . . 88 4 Digits of Numbers 91 4.1 The last digits of a number . . . . . . . . . . . . . . . . . . 91 4.2 The sum of the digits of a number . . . . . . . . . . . . . . 94 4.3 Other problems involving digits . . . . . . . . . . . . . . . . 100 5 Basic Principles in Numb er Theory 103 5.1 Two simple principles . . . . . . . . . . . . . . . . . . . . . 103 5.1.1 Extremal arguments . . . . . . . . . . . . . . . . . . 103 5.1.2 Pigeonhole principle . . . . . . . . . . . . . . . . . . 105 5.2 Mathematical induction . . . . . . . . . . . . . . . . . . . . 108 5.3 Infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.4 Inclusion-exclusion . . . . . . . . . . . . . . . . . . . . . . . 115 6 Arithmetic Functions 119 6.1 Multiplicative functions . . . . . . . . . . . . . . . . . . . . 119 6.2 Number of divisors . . . . . . . . . . . . . . . . . . . . . . . 126 6.3 Sum of divisors . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.4 Euler’s totient function . . . . . . . . . . . . . . . . . . . . . 131 6.5 Exponent of a prime and Legendre’s formula . . . . . . . . 135 7 More on Divisibility 141 7.1 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . 141 7.2 Euler’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 147 7.3 The order of an element . . . . . . . . . . . . . . . . . . . . 150 7.4 Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 153 8 Diophantine Equations 157 8.1 Linear Diophantine equations . . . . . . . . . . . . . . . . . 15 7 8.2 Quadratic Diophantine equations . . . . . . . . . . . . . . . 161 Contents 3 8.2.1 Pythagorean equation . . . . . . . . . . . . . . . . . 161 8.2.2 Pell’s equation . . . . . . . . . . . . . . . . . . . . . 164 8.2.3 Other quadratic equations . . . . . . . . . . . . . . . 169 8.3 Nonstandard Diophantine equations . . . . . . . . . . . . . 171 8.3.1 Cubic equations . . . . . . . . . . . . . . . . . . . . . 171 8.3.2 High-order polynomial equations . . . . . . . . . . . 173 8.3.3 Exponential Diophantine equations . . . . . . . . . . 176 9 Some spe cial problems in number theory 179 9.1 Quadratic residues. Legendre’s symbol . . . . . . . . . . . . 179 9.2 Special numbers . . . . . . . . . . . . . . . . . . . . . . . . 188 9.2.1 Fermat’s numbers . . . . . . . . . . . . . . . . . . . . 188 9.2.2 Mersenne’s numbers . . . . . . . . . . . . . . . . . . 191 9.2.3 Perfect numbers . . . . . . . . . . . . . . . . . . . . . 192 9.3 Sequences of integers . . . . . . . . . . . . . . . . . . . . . . 193 9.3.1 Fibonacci and Lucas sequences . . . . . . . . . . . . 193 9.3.2 Problems involving linear recursive relations . . . . . 197 9.3.3 Nonstandard sequences of integers . . . . . . . . . . 204 10 Problems Involving Binomial Coefficients 211 10.1 Binomial coefficients . . . . . . . . . . . . . . . . . . . . . . 211 10.2 Lucas’ and Kummer’s Theorems . . . . . . . . . . . . . . . 218 11 Miscel laneous Problems 223 II SOLUTIONS TO PROPOSED PROBLEMS 229 12 Divisibility 231 12.1 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 12.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . 237 12.3 The gre atest c ommon divisor . . . . . . . . . . . . . . . . . 242 12.4 Odd and even . . . . . . . . . . . . . . . . . . . . . . . . . . 247 12.5 Modular arithmetics . . . . . . . . . . . . . . . . . . . . . . 248 12.6 Chinese rema inder theorem . . . . . . . . . . . . . . . . . . 251 12.7 Numerical systems . . . . . . . . . . . . . . . . . . . . . . . 253 13 Powers of Integers 261 13.1 Perfect squares . . . . . . . . . . . . . . . . . . . . . . . . . 261 4 Contents 13.2 Perfect cubes . . . . . . . . . . . . . . . . . . . . . . . . . . 270 13.3 k th powers of integers, k ≥ 4 . . . . . . . . . . . . . . . . . . 2 72 14 Floor Function and Fractional Part 275 14.1 General problems . . . . . . . . . . . . . . . . . . . . . . . . 275 14.2 Floor function and integer points . . . . . . . . . . . . . . . 279 14.3 An useful result . . . . . . . . . . . . . . . . . . . . . . . . . 280 15 Digits of Numbers 283 15.1 The last digits of a number . . . . . . . . . . . . . . . . . . 283 15.2 The sum of the digits of a number . . . . . . . . . . . . . . 284 15.3 Other problems involving digits . . . . . . . . . . . . . . . . 288 16 Basic Principles in Number Theory 291 16.1 Two simple principles . . . . . . . . . . . . . . . . . . . . . 291 16.2 Mathematical induction . . . . . . . . . . . . . . . . . . . . 294 16.3 Infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . 300 16.4 Inclusion-exclus ion . . . . . . . . . . . . . . . . . . . . . . . 301 17 Arithmetic Functions 305 17.1 Multiplicative functions . . . . . . . . . . . . . . . . . . . . 305 17.2 Number of divisors . . . . . . . . . . . . . . . . . . . . . . . 307 17.3 Sum of divisors . . . . . . . . . . . . . . . . . . . . . . . . . 309 17.4 Euler’s totient function . . . . . . . . . . . . . . . . . . . . . 311 17.5 Exponent of a prime and Legendre’s formula . . . . . . . . 3 13 18 More on Divisibility 319 18.1 Fermat’s Little Theorem . . . . . . . . . . . . . . . . . . . . 319 18.2 Euler’s Theor e m . . . . . . . . . . . . . . . . . . . . . . . . 326 18.3 The order of an element . . . . . . . . . . . . . . . . . . . . 328 18.4 Wilson’s Theorem . . . . . . . . . . . . . . . . . . . . . . . 330 19 Diophantine Equations 333 19.1 Linear Diophantine equations . . . . . . . . . . . . . . . . . 3 33 19.2 Quadr atic Diophantine equations . . . . . . . . . . . . . . . 336 19.2.1 Pythagorean equations . . . . . . . . . . . . . . . . 336 19.2.2 Pell’s equation . . . . . . . . . . . . . . . . . . . . . 337 19.2.3 Other quadratic equations . . . . . . . . . . . . . . 340 19.3 Nonstandard Diophantine equations . . . . . . . . . . . . . 343 Contents 5 19.3.1 Cubic equations . . . . . . . . . . . . . . . . . . . . 343 19.3.2 High-order polynomial equations . . . . . . . . . . . 345 19.3.3 Exponential Diophantine equations . . . . . . . . . 347 20 Some special problems in number theory 351 20.1 Quadr atic residues. Legendre’s symbol . . . . . . . . . . . . 351 20.2 Special numbers . . . . . . . . . . . . . . . . . . . . . . . . 354 20.2.1 Fermat’s numbers . . . . . . . . . . . . . . . . . . . 354 20.2.2 Mersenne’s numbers . . . . . . . . . . . . . . . . . . 356 20.2.3 Perfect numbers . . . . . . . . . . . . . . . . . . . . 3 57 20.3 Sequences of integers . . . . . . . . . . . . . . . . . . . . . . 357 20.3.1 Fibonacci and Lucas sequences . . . . . . . . . . . . 357 20.3.2 Problems involving linear recursive relations . . . . 360 20.3.3 Nonstandar d sequences of integers . . . . . . . . . . 364 21 Problems Involving Binomial Coefficients 379 21.1 Binomial coefficients . . . . . . . . . . . . . . . . . . . . . . 379 21.2 Lucas’ and Kummer’s Theorems . . . . . . . . . . . . . . . 384 22 Miscel laneous Problems 387 Glossary 393 References 401 Index of Authors 407 Subje ct Index 409 6 Contents [...]... Prime Number Theorem independently of Hadamard in 1896 4 Paul Erd¨s (1913-1996), one of the greatest mathematician of the 20th century o Erd¨s posed and solved problems in number theory and other areas and founded the o field of discrete mathematics 5 Atle Selberg (1917- ), Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms 1.2 PRIME NUMBERS...Foreword One of the oldest and liveliest branches of mathematics, Number Theory, is noted for its theoretical depth and applications to other fields, including representation theory, physics, and cryptography The forefront of Number Theory is replete with sophisticated and famous open problems; at its foundation, however, are basic, elementary ideas that can stimulate and challenge beginning students... unique and vast experience of the authors It captures the spirit of an important mathematical literature and distills the essence of a rich problem-solving culture Number Theory: Structures, Examples and Problems will appeal to senior high school and undergraduate students, their instructors, as well as to all who would like to expand their mathematical horizons It is a source of fascinating problems. .. as it can be easily seen by expanding the brackets The number n has (a + 1)(b + 1) positive divisors and their arithmetic mean is M= (1 + p + p2 + · · · + pa )(1 + q + q 2 + · · · + q b ) (a + 1)(b + 1) If p and q are both odd numbers, we can take a = p and b = q, and it is easy to see that m is an integer 1.2 PRIME NUMBERS 27 If p = 2 and q odd, choose again b = q and consider a + 1 = 1 + q + q... only consecutive squares are 0 and 1 Now assume p is odd We first rule out the case where k is divisible by p: if k = np, then k 2 − pk = p2 n(n − 1), and n and n − 1 are consecutive numbers, so they cannot both be squares We thus assume k and p are coprime, in which case k and k − p are coprime Thus k 2 − pk is a square if and only if k and k − p are squares, say k = m2 and k − p = n2 Then p = m2 −... divisibility theorems and Diophantine equations Emphasis is also placed on the presentation of some special problems involving quadratic residues, Fermat, Mersenne, and perfect numbers, as well as famous sequences of integers such as Fibonacci, Lucas, and other important ones defined by recursive relations By thoroughly discussing interesting examples and applications and by introducing and illustrating... infer that a divides b Any number n that ends in 0 is 1.1 DIVISIBILITY 19 therefore a solution If b = 0, then a is a digit and n is one of the numbers 11, 12, , 19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 56, 77, 88 or 99 Problem 1.1.7 Find the greatest positive integer x such that 236+x divides 2000! Solution The number 23 is prime and divides every 23rd number In 2000 = 86 numbers from 1 to 2000 that... Analytic Number Theory showing that lim n→∞ π(n) n = 1, log n where π(n) denotes the number of primes ≤ n The relation above is known as the Prime Number Theorem It was proved by Hadamard2 and de la Vall´e Poussin3 in 1896 An elementary, but difficult proof, was given by e Erd¨s4 and Selberg5 o 2 Jacques Salomon Hadamard (1865-1963), French mathematician whose most important result is the Prime Number. .. rational numbers the set of real numbers the set of positive real numbers the set of nonnegative real numbers the set of n-tuples of real numbers the set of complex numbers the number of elements in the set A A is a proper subset of B A is a subset of B A without B (set difference) the intersection of sets A and B the union of sets A and B the element a belongs to the set A 12 Notation n|m gcd(m, n)... 1.2 PRIME NUMBERS 25 The most important open problems in Number Theory involve primes The recent book of David Wells [Prime Numbers: The Most Mysterious Figures in Maths, John Wiley and Sons, 2005] contains just few of them We mention here only three such open problems: √ √ 1) Consider the sequence (An )n≥1 , An = pn+1 − pn , where pn denotes the nth prime Andrica’s Conjecture states that the following . numbers R + the set of positive real numbers R 0 the set of nonnegative rea l number s R n the set of n-tuples of real number s C the set of complex numbers |A| the number of elements in the set A A. function p k n p k fully divides n f n Fermat’s number M n Mersenne’s number  a p  Legendre’s symbol F n Fibo nacci’s number L n Lucas’ number P n Pell’s number  n k  binomial coefficient Part I STRUCTURES, EXAMPLES, AND. mathematics, Number The- ory, is noted for its theoretical depth and applications to other fields, in- cluding representation theory, physics, and cryptogra phy. The forefront of Number Theory is