Partitions involving D.H. Lehmer numbers

Abstract

Let n ≥ 2 be a fixed integer, let q and c be two integers with q > n and (n, q) = (c, q) = 1. For every positive integer a which is coprime with q we denote by ${\overline{a}_{c}}$ the unique integer satisfying ${1\leq\overline{a}_{c} \leq{q}}$ and ${a\overline{a}_{c} \equiv{c}({\rm mod}\, q)}$. Put

$$L(q)=\{a\in{Z^{+}}: (a,q)=1, n {\not\hskip0.1mm|} a+\overline{a}_{c} \}.$$

The elements of L(q) are called D.H. Lehmer numbers. The main purpose of this paper is to prove that (under natural conditions) every sufficiently large integer can be expressed as the sum of three D.H. Lehmer numbers.