Reversible primes

TL;DR

This paper investigates properties of n-bit integers and their digit reversals, providing bounds on primes, prime factors, and squarefree numbers, along with exponential sum estimates related to these reversals.

Contribution

It introduces new bounds and asymptotic formulas for primes, prime factors, and squarefree integers with digit reversal properties, using sieve and exponential sum techniques.

Findings

Upper bound on expected primes where p and reversed p are prime

Lower bound on integers with bounded prime factors and their reversals

Asymptotic count of n-bit integers with both number and reversal being squarefree

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 $$ \sum_{a \in \mathcal{B}_n} \exp\left(2\pi i (\alpha a + \vartheta \overleftarrow{a})\right) \quad(\alpha,\vartheta \in\mathbb{R}). $$