On the Diophantine Equation n x + 5 y = z 2

In this paper, we study the Diophantine equation n x + 5 y = z 2 , where n is a positive integer and x, y, z are non-negative integers. We found that if n ≡ 1 (mod 4), then the Diophantine equation has no non-negative integer solution. If n ≡ 3 (mod 20) or n ≡ 7 (mod 20), then the Diophantine equation has all non-negative integer solutions, which are (n, x, y, z) = (n, 1, 0, (n+1)0.5), where (n+1)0.5 is a positive integer.