We count the number
S(
x) of quadruples
\( {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} \) for which
$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $
is a prime number and satisfying the determinant condition:
x 1 x 4???
x 2 x 3?=?1. By means of the sieve, one shows easily the upper bound
S(
x)???
x/log
x. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is
S(
x)???
x/log
x.