Elementary Methods in Number Theory Melvyn B. Nathanson Springer [...]... lexicographic order is a total order on the set of k-tuples of positive integers 26 Prove that Nk with the lexicographic order satisfies the following minimum principle: Every nonempty set of k-tuples of positive integers contains a smallest element 1.2 Greatest Common Divisors Algebra is a natural language to describe many results in elementary number theory Let G be a nonempty set, and let G × G denote... is a polynomial with coefficients in the ring R, then N0 (f ) denotes the number of distinct zeros of f (t) in R We denote by Mn (R) the ring of n × n matrices with coefficients in R In the study of Liouville’s method, we use the symbol {f ( )}n= 2 = 0 if n is not a square, f ( ) if n = 2 , ≥ 0 Contents Preface vii Notation and conventions I xi A First Course in Number Theory 1 Divisibility and Primes... mk and obtain n = ak mk + r, where 0 ≤ r < mk Then 0 < mk − r ≤ n − r = ak mk ≤ n < mk+1 Dividing this inequality by mk , we obtain 0 < ak < m Since m and ak are integers, it follows that 1 ≤ ak ≤ m − 1 If r = 0, then n = ak mk is an m-adic representation If r ≥ 1, then mk ≤ r < mk +1 for some nonnegative integer k ≤ k − 1 By the induction assumption, S(k ) is true and r has a unique m-adic representation... · · + n) for all positive integers n, that is, the sum of the cubes of the first n integers is equal to the square of the sum of the first n integers 18 Prove that the principle of mathematical induction is equivalent to the minimum principle 19 Let a and d be integers with d ≥ 1 Prove that there exist unique integers q and r such that a = dq + r and − d d . collaborator for 25 years, and a master of elementary methods in number theory. Preface Arithmetic is where numbers run across your mind looking for the answer. Arithmetic is like numbers spinning. classical theorems of Chebyshev and Mertens on the distribu- tion of prime numbers. Finally, we give elementary proofs of two of the most famous results in mathematics, the prime number theorem, which. Elementary Methods in Number Theory Melvyn B. Nathanson Springer