On a conjecture of Ram\'ırez Alfons\'ın and Skałba III

Abstract

Let $1<c<d$ be two relatively prime integers. For a non-negative integer $\ell$, let $g_{\ell }(c,d)$ be the largest integer $n$ such that $n=cx+dy$ has at most $\ell$ non-negative solutions $(x,y)$. In this paper we prove that $${\pi }_{\ell ,c,d}\sim \frac{\pi (g_{\ell }(c,d))}{2\ell +2}(\text{as}c\to \infty ),$$ where ${\pi }_{\ell ,c,d}$ is the number of primes $n$ having more than $\ell$ distinct non-negative solutions to $n=cx+dy$ with $n\le g_{\ell }(c,d)$, and $\pi (x)$ denotes the number of all primes up to $x$. The case where $\ell =0$ has been proved by Ding, Zhai and Zhao recently, which was conjectured formerly by Ram\'ırez Alfons\'ın and Skałba.