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.

Keywords: Prime, Gauss sums, Dirichlet character of polynomials.