Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Prime Divisors
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915284905885696 |
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| author | Xie, Likun |
| author_facet | Xie, Likun |
| contents | We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^θ$ for some $θ> 0$ depending on $k$. The proof uses a variant of Chen's method, weighted sieves, and Elliott's results on primes in arithmetic progressions with large power-factor moduli. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_14133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Prime Divisors Xie, Likun Number Theory We prove lower bounds for the number of primes $p \leq N + b$ such that $p-b$ is divisible by $2^{k(N)}$ and has at most $k$ odd prime factors ($k \geq 2$), assuming $2^{k(N)} \leq N^θ$ for some $θ> 0$ depending on $k$. The proof uses a variant of Chen's method, weighted sieves, and Elliott's results on primes in arithmetic progressions with large power-factor moduli. |
| title | Primes $p$ such that $p-b$ Has a Large Power Factor and Few Other Prime Divisors |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.14133 |