Let \(K = \mathbb {Q}(\sqrt{d})\) be a quadratic field, and let \(\mathcal {O}_{K}\) denote its ring of integers. In this work, we investigate the solvability of the Diophantine equation \(r + s + t = rst = n\) in finite fields \(\mathbb {F}_{p^k}\) , where p is an odd prime and k is a natural number, as well as in the ring of integers \(\mathcal {O}_{K}\) of the quadratic field \(K = \mathbb {Q}(\sqrt{d})\) . We explicitly solve the system for certain initial values of n, thereby corroborating our theoretical results.
