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

## Authors

• Wachirarak Orosram Department of Mathematics Faculty of Science Buriram Rajabhat University

## Keywords:

Diophantine equation, Catalan's conjecture

## 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}).

2023-10-30

## How to Cite

Orosram, W. (2023). On the Diophantine Equation (p+n)^x+p^y=z^2 where p and p+n are Prime Numbers. European Journal of Pure and Applied Mathematics, 16(4), 2208–2212. https://doi.org/10.29020/nybg.ejpam.v16i4.4822

## Section

Nonlinear Analysis