An additive problem in connection with some quadratic real fields with class number one

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 $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.