2025/06/04 by Chen, Yong-Gao, Zhu, Hui · 1 citation
#11D07 #11N13 #11Y35 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2506.03620
For two coprime positive integers a,b, let T(a,b)=\ ax+by : x,y∈ ℤ≥ 0 \ and let s(a,b)=ab-a-b. It is well known that all integers which are greater than s(a,b) are in T(a,b). Let π(a, b) be the number of primes in T(a,b) which are less than or equal to s(a,b). It is easy to see that π(2, 3)=0 and π(2, b)=1 for all odd integers b≥ 5. In this paper, we prove that if b>a≥ 3 with gcd (a, b)=1, then π(a, b)>0.005 s(a,b)/log s(a,b). We conjecture that (13)/(66)π(s(a,b))≤ π(a, b)≤ \frac 12π(s(a,b)) for all b>a≥ 3 with gcd (a, b)=1.