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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.11380 |
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| _version_ | 1866916158086578176 |
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| author | Dartyge, Cécile Martin, Bruno Rivat, Joël Shparlinski, Igor E. Swaenepoel, Cathy |
| author_facet | Dartyge, Cécile Martin, Bruno Rivat, Joël Shparlinski, Igor E. Swaenepoel, Cathy |
| contents | For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j}, $$ the digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a \text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$ \left|\{a \in \mathcal{B}_n:~ \max \{Ω(a), Ω(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ where $Ω(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2πi (αa + \vartheta \overleftarrow{a})\right) \quad(α,\vartheta \in\mathbb{R}). $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11380 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reversible primes Dartyge, Cécile Martin, Bruno Rivat, Joël Shparlinski, Igor E. Swaenepoel, Cathy Number Theory 11A63, 11N05, 11N36 For an $n$-bit positive integer $a$ written in binary as $$ a = \sum_{j=0}^{n-1} \varepsilon_{j}(a) \,2^j $$ where, $\varepsilon_j(a) \in \{0,1\}$, $j\in\{0, \ldots, n-1\}$, $\varepsilon_{n-1}(a)=1$, let us define $$ \overleftarrow{a} = \sum_{j=0}^{n-1} \varepsilon_j(a)\,2^{n-1-j}, $$ the digital reversal of $a$. Also let $\mathcal{B}_n = \{2^{n-1}\leq a<2^n:~a \text{ odd}\}.$ With a sieve argument, we obtain an upper bound of the expected order of magnitude for the number of $p \in \mathcal{B}_n$ such that $p$ and $\overleftarrow{p}$ are prime. We also prove that for sufficiently large $n$, $$ \left|\{a \in \mathcal{B}_n:~ \max \{Ω(a), Ω(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ where $Ω(n)$ denotes the number of prime factors counted with multiplicity of $n$ and $c > 0$ is an absolute constant. Finally, we provide an asymptotic formula for the number of $n$-bit integers $a$ such that $a$ and $\overleftarrow{a}$ are both squarefree. Our method leads us to provide various estimates for the exponential sum $$ \sum_{a \in \mathcal{B}_n} \exp\left(2πi (αa + \vartheta \overleftarrow{a})\right) \quad(α,\vartheta \in\mathbb{R}). $$ |
| title | Reversible primes |
| topic | Number Theory 11A63, 11N05, 11N36 |
| url | https://arxiv.org/abs/2309.11380 |