A Cubic Analogue of the Jacobsthal Identity

It is well known that if p is a prime such that p ≡ 1 (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 = x2 + y2, 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 ≡ 1 (mod 6), then 3p = u2 + uv + v2 for some integers u and v using an analogue of Jacobsthal’s identity.