Some special primes and square sums

Abstract

The main purpose of this paper is using the properties of Gauss sums and character sums to study a classical problem related to prime p with p ≡ 1(mod 3), and prove an interesting result. That is, we shall prove that if p ≡ 1(mod 3), then there exist three integers x, y and z such that (x³ − y³)² + (y³ − z³)² + (z³ − x³)² = 162p³. Moreover, we also give an exact representation for integers x, y and z.