English

Prime numbers with an almost prime reverse

Number Theory 2025-07-11 v2

Abstract

Let bb be an integer greater than or equal to 22. For any integer n[bλ1,bλ1]n\in \left[b^{\lambda-1}, b^{\lambda}-1\right], we denote by Rλ(n)R_\lambda (n) the reverse of nn in base bb, obtained by reversing the order of the digits of nn. 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 ΩbN\Omega_b\in\mathbb{N} and cb>0c_b>0 such that, for at least cbbλλ2c_b b^{\lambda} \lambda ^{-2} primes p[bλ1,bλ1]p\in \left[b^{\lambda-1}, b^{\lambda}-1\right], the reverse Rλ(p)R_\lambda(p) has at most Ωb\Omega_b prime factors. Some explicit admissible values of Ωb\Omega_b are given.

Keywords

Cite

@article{arxiv.2506.21642,
  title  = {Prime numbers with an almost prime reverse},
  author = {Cécile Dartyge and Joël Rivat and Cathy Swaenepoel},
  journal= {arXiv preprint arXiv:2506.21642},
  year   = {2025}
}

Comments

Minor corrections and page settings modified. 92 pages

R2 v1 2026-07-01T03:35:15.166Z