The number of solutions of a certain quadratic congruence related to the class number of 𝑄(√𝑝)

Abstract

Let p p be an odd prime, and let k k be a positive integer with i ⩽ k ⩽ ( p − 1 ) / 2 i \leqslant k \leqslant (p - 1)/2 . In this note we give a formula for the number of solutions ( x 1 , … , x k ) ({x_1}, \ldots ,{x_k}) of the congruence x 1 2 + ⋯ + x k 2 ≡ 0 ( mod p ) x_1^2 + \cdots + x_k^2 \equiv 0\;(\bmod p) , 1 ⩽ x 1 > ⋯ > x k ⩽ ( p − 1 ) / 2 1 \leqslant {x_1} > \cdots > {x_k} \leqslant (p - 1)/2 .