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Journal Articles Journal of the London Mathematical Society Year : 2024

## Reversible primes

Cécile Dartyge
• Function : Author
• PersonId : 937256
Bruno Martin
Joël Rivat
Igor E. Shparlinski
• Function : Author
• PersonId : 1345194
Cathy Swaenepoel

#### Abstract

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 $n$, $\left|\{a \in \mathcal{B}_n:~ \max \{\Omega (a), \Omega (\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2},$ where $\Omega(n)$ 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 $\sum_{a \in \mathcal{B}_n} \exp\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right) \quad(\alpha,\vartheta \in\mathbb{R}).$

#### Domains

Mathematics [math] Number Theory [math.NT]

### Dates and versions

hal-04430339 , version 1 (31-01-2024)

### Identifiers

• HAL Id : hal-04430339 , version 1
• ARXIV :

### Cite

Cécile Dartyge, Bruno Martin, Joël Rivat, Igor E. Shparlinski, Cathy Swaenepoel. Reversible primes. Journal of the London Mathematical Society, 2024, 109 (3), pp.e12883. ⟨hal-04430339⟩

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