Prime numbers with an almost prime reverse

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.