Solutions of the Diophantine Equation 2x + py = z2 When p is Prime

Abstract

In this article, we consider the Diophantine equation 2 x + p y = z 2 when p = 4N + 3 and p = 4N + 1 are primes. The values x, y, z are positive integers. For each prime, all the possibilities for solutions are investigated. All cases of no-solutions, as well as cases of infinitely many solutions are determined. Whenever the number of solutions for p = 4N + 3 / p = 4N + 1 is finite, we establish the respective connection between this number to all Mersenne Primes / Fermat Primes known as of 2018. Numerical solutions of various cases are also exhibited.

Prime Solutions to the Diophantine Equation 2ⁿ = p² +7

“…We take a tour of prime numbers in the next sections, discussing their importance in relation to Diophantine equations and revealing the amazing answers that result from our investigation. By clarifying the underlying ideas that govern the relationship between primes and powers of two, we want to improve our understanding of these core ideas in mathematics and push the frontiers of what is known about it [1][2][3].…”

On the Non-Linear Diophantine Equation 61^x + 67^y = z^2 and 67^x + 73^y = z^2

“…In 2018, Burshtein [2] discussed on an open problem of Chotchaisthit, on the Diophantine equation 2 x + p y = z 2 , where p are particular prime and y=1. In 2018, Burshtein [4] also discussed on the Diophantine equation 2 x + p y = z 2 , where p are prime. In this paper we consider some particular exponential Diophantine equations 61 x + 67 y = z 2 (1) and 67 x + 73 y = z 2 (2) where x, y , and z are non-negative integers.…”

On the Non-Linear Diophantine Equations 31x + 41y = z2 and 61x + 71y = z2

“…In [2], Burshtein discussed an open problem of Chotchaisthit, on the Diophantine equation 2 x + p y = z 2 , where p is particular prime and y = 1. In [4], Burshtein also discussed on the Diophantine equation 2 x + p y = z 2 for odd prime p and x, y and z are positive integers. In [6], S. Kumar et.al.…”