Class numbers, Ono invariants and some interesting primes

Abstract

Our aim is to find all the prime numbers $p$ such that $p+x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2\le p$. We solve entirely the cases $p\equiv 1,3,5(mod8)$, using the knowledge of the quadratic imaginary number fields with class numbers 4, 1 and 2 respectively. The case $p\equiv 7(mod8)$ is not completely solved. Taking into account a result of Stéphane Louboutin, we prove that there is at most one value $p\equiv 7(mod8)$ besides our list. Assuming a Restricted Riemann Hypothesis, the list is complete. In the last section of the paper we give a short sketch for the general problem: find all odd integers $n>1$ such that $n+x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2\le n$.