Class numbers, Ono invariants and some interesting primes

Abstract

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod 8$. The idea of the proof in these cases is to find the class number for the quadratic imaginary field $\mathbb{Q}(i\sqrt p)$. Since we know all these quadratic imaginary fields with class number 1, 2 or 4, we are able to solve these cases. The most interesting case is $p\equiv 7 \pmod 8$. We prove in this case that the Ono invariant of the field equals the class number. S. Louboutin succeeded to find all these fields, with one possible exception. Assuming a Restricted Riemann Hypothesis, the list of Louboutin is complete.