Primes and polygonal numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912528734355456 |
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| author | Bhattacharya, Soumya Rahaman, Habibur |
| author_facet | Bhattacharya, Soumya Rahaman, Habibur |
| contents | A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the natural numbers, whereas the sum of the reciprocals of such primes converges unless $T_r(m)=m^2$ and $T_s(n)=n^2$. For each pair of coprime positive integers $a$ and $b$, there are arbitrary long arithmetic progressions among the primes of the form $am^2+bn^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13650 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primes and polygonal numbers Bhattacharya, Soumya Rahaman, Habibur Number Theory 11N32, 11N13, 11N36 (Primary) 11P32, 11B25, 11B05 (Secondary) A linear combination $aT_r(m)+bT_s(n)$ of an \mbox{$r$-gonal} number $T_r(m)$ and an $s$-gonal number $T_s(n)$ with mutually coprime positive integer coefficients $a$ and $b$ produces infinitely many primes as $m$ and~$n$ varies over the natural numbers, whereas the sum of the reciprocals of such primes converges unless $T_r(m)=m^2$ and $T_s(n)=n^2$. For each pair of coprime positive integers $a$ and $b$, there are arbitrary long arithmetic progressions among the primes of the form $am^2+bn^2$. |
| title | Primes and polygonal numbers |
| topic | Number Theory 11N32, 11N13, 11N36 (Primary) 11P32, 11B25, 11B05 (Secondary) |
| url | https://arxiv.org/abs/2408.13650 |