Publication: On The Solvability Of Some Diophantine Equations Of The Form ax+by = z2
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Date
2022-07
Authors
Amr Moustafa Mohamed Aly, Elsayed Elshahed
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Abstract
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.
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Keywords
Diophantine Equations , ax+by = z2