Primes of the form $ax+by$ in certain intervals with small solutions

Let $1<a<b$ be two relatively prime integers and Z≥ 0 the set of non-negative integers. For any non-negative integer , denote by g,a,b the largest integer n such that the equation n=ax+by, (x,y)≥ 0² (1) has at most solutions. Let π,a,b be the number of primes p≤ g,a,b having at least +1 solutions for (1) and $π(x)$ the number of primes not exceeding x. In this article, we prove that for a fixed integer a≥ 3 with (a,b)=1, π,a,b=(a-2/2( a+a-1)+o(1))π(g,a,b)(as~ b→∞). For any non-negative and relatively prime integers $a,b$, satisfying e⁺¹≤ a<b, we show that {equation*} π,a,b>0.005· 1/+1{g,a,b}{log g,a,b}. {equation*} Let π,a,b* be the number of primes p≤ g,a,b having at most solutions for (1). For an integer a≥ 3 and a large sufficiently integer b with (a,b)=1, we also prove that π*,a,b>(2+1)a/2( a+a-1)g,a,blog g,a,b. Moreover, if <a<b with (a,b)=1, then we have {equation*} π*,a,b>+0.02/+1{g,a,b}{log g,a,b}. {equation*} These results generalize the previous ones of Chen and Zhu (2025), who established the results for the case =0.