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In this paper we prove that $$ \\pi_{\\ell,c,d}\\sim\\frac{\\pi\\bigl(g_\\ell(c,d)\\bigr)}{2 \\ell+2}\\quad(\\text{as}~ c\\to\\infty)\\,, $$ where $\\pi_{\\ell,c,d}$ is the number of primes $n$ having more than $\\ell$ distinct non-negative solutions to $n=c x+d y$ with $n\\le g_\\ell(c,d)$, and $\\pi(x)$ denotes the number of all primes up to $x$. 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For a non-negative integer $\\ell$, let $g_\\ell(c,d)$ be the largest integer $n$ such that $n=c x+d y$ has at most $\\ell$ non-negative solutions $(x,y)$. In this paper we prove that $$ \\pi_{\\ell,c,d}\\sim\\frac{\\pi\\bigl(g_\\ell(c,d)\\bigr)}{2 \\ell+2}\\quad(\\text{as}~ c\\to\\infty)\\,, $$ where $\\pi_{\\ell,c,d}$ is the number of primes $n$ having more than $\\ell$ distinct non-negative solutions to $n=c x+d y$ with $n\\le g_\\ell(c,d)$, and $\\pi(x)$ denotes the number of all primes up to $x$. 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