Primes of the form $ax+by$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913874944458752 |
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| author | Chen, Yong-Gao Zhu, Hui |
| author_facet | Chen, Yong-Gao Zhu, Hui |
| contents | For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 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 $b\ge 5$. In this paper, we prove that if $b>a\ge 3$ with $\gcd (a, b)=1$, then $π(a, b)>0.005 s(a,b)/\log s(a,b)$. We conjecture that $\frac{13}{66}π(s(a,b))\le π(a, b)\le \frac 12π(s(a,b))$ for all $b>a\ge 3$ with $\gcd (a, b)=1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Primes of the form $ax+by$ Chen, Yong-Gao Zhu, Hui Number Theory 11D07, 11N13, 11Y35 For two coprime positive integers $a,b$, let $T(a,b)=\{ ax+by : x,y\in \mathbb{Z}_{\ge 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 $b\ge 5$. In this paper, we prove that if $b>a\ge 3$ with $\gcd (a, b)=1$, then $π(a, b)>0.005 s(a,b)/\log s(a,b)$. We conjecture that $\frac{13}{66}π(s(a,b))\le π(a, b)\le \frac 12π(s(a,b))$ for all $b>a\ge 3$ with $\gcd (a, b)=1$. |
| title | Primes of the form $ax+by$ |
| topic | Number Theory 11D07, 11N13, 11Y35 |
| url | https://arxiv.org/abs/2506.03620 |