Some identities involving primes

Abstract

The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study some classical problem related to a prime p. We shall give some new identities involving a prime p, one of which is that if prime p = 6k + 1, then there exist three positive integers x, y and z such that p² = x² +y² +z². At the same time, we also give an exact expression for x, y and z.

Mathematics Subject Classification (2010): 11L05, 11P32.
Key words: Gauss sums, the equation related to primes, solutions, analytic method.