Reversible primes

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 $n$, $$ \left|\{a \in \mathcal{B}_n:~ \max \{Ω(a), Ω(\overleftarrow{a})\}\le 8 \}\right| \ge c\, \frac{2^n}{n^2}, $$ 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 $$ \sum_{a \in \mathcal{B}_n} \exp\left(2πi (αa + \vartheta \overleftarrow{a})\right) \quad(α,\vartheta \in\mathbb{R}). $$

Figures (1)

• Figure 1: Base 2: graph of $\varTheta(n)/\varTheta_{\text{exp}}(n)$ for $n\leqslant 50$.

Theorems & Definitions (39)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Lemma 1.5
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof