Prime numbers with an almost prime reverse

TL;DR

This paper proves that a positive proportion of prime numbers have reverses with a bounded number of prime factors, using advanced sieve methods and a Bombieri-Vinogradov type theorem for reversed primes.

Contribution

It establishes a Bombieri-Vinogradov type theorem for reversed primes and demonstrates the existence of primes whose reverses have a bounded number of prime factors.

Findings

A positive proportion of primes have reverses with at most _b prime factors.

Explicit bounds for the maximum number of prime factors in reversed primes are provided.

The results apply to digits in any base b2 and involve advanced sieve techniques.

Abstract

Let $b$ be an integer greater than or equal to $2$. For any integer $n\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, we denote by $R_\lambda (n)$ 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 $\Omega_b\in\mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^{\lambda} \lambda ^{-2}$ primes $p\in \left[b^{\lambda-1}, b^{\lambda}-1\right]$, the reverse $R_\lambda(p)$ has at most $\Omega_b$ prime factors. Some explicit admissible values of $\Omega_b$ are given.