Abstract
In this paper, we study the Diophantine equation x2 = n2 + mn + np + 2mp with m, n, p, and x being natural numbers. This equation arises from a geometry problem and it leads to representations of primes by each of the three quadratic forms: a2 + b2, a2 + 2b2, and 2a2 − b2. We show that there are infinitely many solutions and conjecture that there are always solutions if x ≥ 5 and x ≠ 7; and, we find a parametrization of the solutions in terms of four integer variables.