On the Diophantine Equation (p+n)^x+p^y=z^2 where p and p+n are Prime Numbers

Abstract

In this paper, we study the Diophantine equation (p+n)^x+p^y=z^2, where p, p+n are prime numbers and n is a positive integer such that n equiv mod 4. In case p=3 and n=4, Rao{7} showed that the non-negative integer solutions are (x,y,z)=(0,1,2) and (1,2,4) In case p>3 and pequiv 3pmod4, if n-1 is a prime number and 2n-1 is not prime number, then the non-negative integer solution (x, y, z) is (0, 1,\sqrt {p+1}) or ( 1, 0, \sqrt{p+n+1}). In case pequiv 1pmod4, the non-negative integer solution (x,y,z) is also (0, 1,\sqrt {p+1}) or ( 1,0, \sqrt{p+n+1}).