Jacobsthal identity for Q(sqrt(-2))

TL;DR

This paper derives explicit formulas for the integers a and b in the representation p=a^2+2b^2 for primes p congruent to 1 or 3 mod 8, using Jacobsthal sums, extending classical identities.

Contribution

It provides new closed-form expressions for the solutions to p=a^2+2b^2 in terms of Jacobsthal sums, generalizing Jacobsthal's classical identity.

Findings

Explicit formulas for a and b in terms of Jacobsthal sums

Extension of classical Jacobsthal identity to primes p ≡ 1 or 3 mod 8

New insights into representations of primes as sums of squares

Abstract

Let $p$ be a prime congruent to 1 or 3 modulo 8 so that the equation $p=a^2+2b^2$ is solvable in integers. In this paper, we obtain closed-form expressions for $a$ and $b$ in terms of Jacobsthal sums. This is analogous to a classical identity of Jacobsthal.