The aim of this paper is to analyze the set of prime numbers
![$p>3$](img17.png)
for which
![$p-a^2$](img18.png)
is four times a prime for any positive odd integer
![$a$](img19.png)
such that
![$a^2<p$](img20.png)
(we consider that 1 is a prime number). We show that for such prime numbers
![$p$](img21.png)
we have
![$p=x^2+4$](img22.png)
, where
![$x$](img23.png)
is a prime number. We compute also the class number
![$h(-4p)$](img24.png)
for the quadratic imaginary field
![$\mathbb{Q} (i\sqrt {p})$](img25.png)
using a famous formula of Gauss. There are only six prime numbers with the above property.