ON THE DIOPHANTINE EQUATION $(p+1)^{2x}+q^y=z^2$

Abstract

In this paper, we found that the Diophantine equation (p + 1) 2x + q y = z 2 has no non-negative integer solution where p is a Mersenne prime number which q − p = 2 and x, y and z are non-negative integers.

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

“…Many mathematicians have been studying the Diophantine equations of the type (p + n) x +p y = z 2 with a constant n and a specific condition of p, for example, in case that p is a prime number. In 2015, Tatong and Suvarnamani [10] found that (p, x, y, z) = (3, 1, 0, 2) is a unique non-negative integer solution of the Diophantine equation p x +(p+1) y = z 2 where p is an odd prime number. In 2018, Burshtein [1] showed that the Diophantine equation p x + (p + 4) y = z 2 when p > 3, p + 4 are primes has no positive integer solution (x, y, z).…”

The Diophantine equation a x ± a y = z n when a is any nonnegative integer

“…In 2011, Suvarnamani [17] considered the Diophantine equation in the form 2 x + p y = z 2 when p is prime. In 2012, Tatong and Suvarnamani [18] studied the Diophantine equation in the form p x + p y = z 2 , where p equals 2 or 3 and x, y, z are nonnegative integers. In 2019, Burshtein [5] generalized the work of Tatong and Suvarnamani and presented the Diophantine equations p x + p y = z 2 and p x − p y = z 2 when p 2 is prime and x, y, z are positive integers.…”