WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 22, 2023
On the Diophantine Equation nx + 10y = z2
Author:
Abstract: In this paper, we show that $$(n, x, y, z) = (2, 3, 0, 3)$$ is the unique non-negative integer solution of the Diophantine equation $$n^{x} + 10^{y} = z^{2}$$, where n is a positive integer with $$n ≡ 2 (mod 30)$$ and $$x, y, z$$ are non-negative integers. If $$n = 5$$, then the Diophantine equation has exactly one non-negative integer solution $$(x, y, z) = (3, 2, 15)$$. We also give some conditions for non-existence of solutions of the Diophantine equation.