On the diophantine equation px+5y=z2

Abstract

The aim of this work is to show that the Diophantine equations p z x y   5 2 where p is prime number and p satisfies; case 1: p 1 (mod 4) or case 2: p  3 (mod4) and p  2 (mod5) or case 3: p  3 (mod4) and p  3 (mod5) , has no non-negative integer solution. In addition, the Diophantine
equations p z x y   5 2 have some non-negative integer solutions (p, x, y, z) if p  3(mod 4) and p 1(mod5) , such as (11, 1, 2, 6), (11, 1, 1, 4), (31, 1, 1, 6), (71, 1, 3, 4), (131, 1, 3, 16) and (191, 1, 1, 14), etc.
Moreover, if p  3(mod 4) and p  4(mod5), such as (19, 1, 3, 12), (139, 1, 1, 12) and (199, 1, 3, 18).