To make the book appropriate for a wide audience, we have included large collections of problems of varying difficulty. Some effort has been devoted to make the first chapters less demanding.

A simple proof that is irrational Ivan Niven Let = a=b, the quotient of positive integers. We de ne the polynomials xn(a bx)n f(x) = ; n!

the high degree b-Niven numbers shown in this paper are given by explicit formulas nd have all digits di erent from zero. In Example 1 we show that the number [(6510)(6510)]6511 is 6511-Niven of degree 15

Definition: A positive integer is called a Niven number if it is divisible by its digital sum. Various articles have appeared concerning digital sums and properties of the set of Niven numbers.

Reducing A(x jk; l; t) to (x t) takes five steps: Let k = t1k3 with t1 the largest integer such that (t1; q) Now you can transfrom the problem of A(x k; l; t) to A(x t1; l; t) Let t1 = k1k2 with k1 largest such that (k1; …

etailed proof of the irrationality of π The proof is due to Ivan Niven (1947) and essential to the proof are Lemmas 2. and 3 due to Charles Hermite (1800’s). us introdu. e some defin. tions. ∞ X wn Definition. …

A positive integer n is called a Niven number (or a Harshad number) if it is divisible by the sum of its decimal digits. For instance, 2007 is a Niven number since 9 divides 2007.