Partitions into two Lehmer numbers (Q985727)

Let \(c,n,q\) be integers with \(q>n\geq 2\) and \(\gcd(cn,q)=1\). For every positive integer \(a\) with \(\gcd(a,q)=1\) denote by \(\text{inv}_{c,q}(a)\) the unique integer \(1\leq\text{inv}_{c,q}(a)<q\) satisfying \(\text{inv}_{c,q}(a)\equiv ca^{-1}(\bmod q)\). The elements of the set \[ {\mathcal L}_{n,c}(q)=\{a\geq 1:\gcd(a,q)=1,n\not\mid a+\text{inv}_{c,q}(a)\} \] are called \((c,n,q)\)-Lehmer numbers. The authors prove an asymptotic formula for the number of representations of an integer as sum of two Lehmer numbers which implies that under some natural restrictions each sufficiently large integer is a sum of two such numbers.