[...]... tackling various number- theoretic problems Here are some basic ideas: (1) an odd number is of the form 2k + 1, for some integer k; (2) an even number is of the form 2m, for some integer m; (3) the sum of two odd numbers is an even number; (4) the sum of two even numbers is an even number; (5) the sum of an odd and even number is an odd number; (6) the product of two odd numbers is an odd number; (7) a... rational numbers the set of positive rational numbers the set of nonnegative rational numbers the set of n-tuples of 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 coefficient of the term x n in the polynomial p(x) xii Abbreviations and Notation Notation for Sets, Logic, and Number. .. divisors in each pair is 4204 Hence the answer is 4204·562 · 4202 = 4202250 Putting the last three examples together gives two interesting results in number theory For a positive integer n denote by τ (n) the number of its divisors It is 18 104 Number Theory Problems clear that τ (n) = 1 d|n Writing τ in this summation form allows us later to discuss it as an example of a multiplicative arithmetic function... five numbers from the set {1, 2, 3, 4, 5, 6, 7} If he told Claudia what the product of the chosen numbers was, that would not be enough information for Claudia to figure out whether the sum of the chosen numbers was even or odd What is the product of the chosen numbers? Solution: The answer is 420 Providing the product of the chosen numbers is equivalent to telling the product of the two unchosen numbers... positive even numbers greater than 2 are composite In other words, 2 is the only even (and the smallest) prime All other primes are odd; that 6 104 Number Theory Problems is, they are not divisible by 2 The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 How many primes are there? Are we really sure that there are infinitely many primes? Please see Theorem 1.3 below A comparison between the number of... remarkable result in the mathematical field of analytic number theory showing that π(n) = 1, n→∞ n/log n lim where π(n) denotes the number of primes ≤ n The relation above is known as the prime number theorem It was proved by Hadamard and de la Vall´ e Poussin e in 1896 An elementary but difficult proof was given by Erd¨ s and Selberg o 1 Foundations of Number Theory 11 G.C.D For a positive integer k we denote... of m, n the number of primes ≤ n number of divisors of n sum of positive divisors of n a and b are congruent modulo m Euler’s totient function order of a modulo m M¨ bius function o base-b representation the sum of digits of n factorial base expansion floor of x celling of x fractional part of x Legendre’s function p k fully divides n Fermat number Mersenne number 1 Foundations of Number Theory Divisibility... representation 3n = (3n−1 − 1) + 3n−1 + (3n−1 + 1) Example 1.3 Let k be an even number Is it possible to write 1 as the sum of the reciprocals of k odd integers? Solution: The answer is negative We approach indirectly Assume that 1= 1 1 + ··· + n1 nk for some odd integers n 1 , , n k ; then clearing denominators we obtain 4 104 Number Theory Problems n 1 · · · n k = s1 + · · · + sk , where si are all odd But... which 3n − 4, 4n − 5, and 5n − 3 are all prime numbers Solution: The sum of the three numbers is an even number, so at least one of them is even The only even prime number is 2 Only 3n − 4 and 5n − 3 can be even Solving the equations 3n − 4 = 2 and 5n − 3 = 2 yields n = 2 and n = 1, respectively It is trivial to check that n = 2 does make all three given numbers prime Example 1.7 [AHSME 1976] If p and... the number of the lockers that remain open after all the students finish their walks? Solution: Note that the ith locker will be operated by student j if and only if j | i By property (g), this can happen if and only if the locker will also be operated by student ij Thus, only the lockers numbered 1 = 12 , 4 = 22 , 9 = 32 , 1 Foundations of Number Theory 3 and 16 = 42 will be operated on an odd number . two odd numbers is an even number; (4) the sum of two even numbers is an even number; (5) the sum of an odd and even number is an odd number; (6) the product of two odd numbers is an odd number; (7). 65 Fermat Numbers 70 Mersenne Numbers 71 Perfect Numbers 72 vi Contents 2 Introductory Problems 75 3 Advanced Problems 83 4 Solutions to Introductory Problems 91 5 Solutions to Advanced Problems. (EB) 104 Number Theory Problems Titu Andreescu, Dorin Andrica, Zuming Feng October 25, 2006 Contents Preface vii Acknowledgments ix Abbreviations and Notation xi 1 Foundations of Number Theory