A Cubic Analogue of the Jacobsthal Identity

Abstract

It is well known that if p is a prime such that p I (mod 4). then p can be expressed as a sum of two squares. Several proofs of this fact are known and one of them, due to E. Jacobsthal, involves the identity p = x(2) + y(2) with x and y expressed explicitly in terms of sums involving the Legendre symbol. These sums are now known as the Jacobsthal sums. In this short note, we prove that if p equivalent to 1 (mod 6). then 3p = u(2) + uv + nu(2) for some integers u and v using an analogue of Jacobsthal's identity.