Prime numbers with an almost prime reverse
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913935577317376 |
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| author | Dartyge, Cécile Rivat, Joël Swaenepoel, Cathy |
| author_facet | Dartyge, Cécile Rivat, Joël Swaenepoel, Cathy |
| contents | Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{λ-1}, b^λ-1\right]$, we denote by $R_λ(n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We establish a Bombieri-Vinogradov type theorem for the set of the reverses of the prime numbers. Combined with sieve methods, this permits us to prove that there exist $Ω_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^λ λ^{-2}$ primes $p\in \left[b^{λ-1}, b^λ-1\right]$, the reverse $R_λ(p)$ has at most $Ω_b$ prime factors. Some explicit admissible values of $Ω_b$ are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime numbers with an almost prime reverse Dartyge, Cécile Rivat, Joël Swaenepoel, Cathy Number Theory 11A63, 11N05, 11N36 Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{λ-1}, b^λ-1\right]$, we denote by $R_λ(n)$ the reverse of $n$ in base $b$, obtained by reversing the order of the digits of $n$. We establish a Bombieri-Vinogradov type theorem for the set of the reverses of the prime numbers. Combined with sieve methods, this permits us to prove that there exist $Ω_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^λ λ^{-2}$ primes $p\in \left[b^{λ-1}, b^λ-1\right]$, the reverse $R_λ(p)$ has at most $Ω_b$ prime factors. Some explicit admissible values of $Ω_b$ are given. |
| title | Prime numbers with an almost prime reverse |
| topic | Number Theory 11A63, 11N05, 11N36 |
| url | https://arxiv.org/abs/2506.21642 |