On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2 - Elshahed, Amr Moustafa Mohamed Aly Elsayed (2022)
The Diophantine equation ax+py = z2 where p is prime is widely studied by many mathematicians. Solving equations of this type often include Catalan’s conjecture in the process of proving these equations. Here, we study the non-negative integer solutions for some Diophantine equations of such family. We will use Mihailescu’s theorem (which is the proof of Catalan’s conjecture) and elementary methods to solve the Diophantine equations 16x −7y = z2, 16x − py = z2 and 64x − py = z2, then we will study a generalization where (4n)x − py = z2 and x, y, z,n are non-negative integers. By using Mihailescu’s theorem and a fundamental approach in the theory of numbers, namely the theory of congruence, we will determine the solution of the Diophantine equations 7x+11y = z2, 13x+17y = z2, 15x+17y = z2 and 2x+257y = z2 where x, y and z are non-negative integers. Also, we will prove that for any non-negative integer n, all non-negative integer solutions of the Diophantine equation 11n8x+11y = z2 are of the form (x, y, z) = (1,n,3(11) n2 ) where n is even, and has no solution when n is odd. Finally, we will concentrate on finding the solutions of the Diophantine equation 3x+ pmny = z2 where y = 1,2 and p > 3 a prime number.