About the number of τ-numbers relative to polynomials with integer coefficients

Abstract

We show that for all polynomials Q(x) with integer coefficients, that satisfy the extra condition |Q(0) · Q(1) | ≠ 1, there are infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). We also find some examples of polynomials Q(x) for which 1 is the only τ-number relative to the polynomial Q(x) and some examples of polynomials Q(x) with |Q(0) · Q(1)|= 1, which have infinitely many positive integers n such that n is a τ-number relative to the polynomial Q(x). In addition, we prove one result about the generators of a τ-number.