Title: 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.