On the Diophantine Equation t^x + (t + 3k)^y = z^2 Where t is a Positive Integer - Jantoy, Darakorn; Intep, Somkid (Burapha Science Journal, 2024)
Background and Objectives : to find the solution to the Diophantine equation tx + (t + 3k)y = z2 where k, x, y and z are non-negative integers, and is a positive integer, which is in the form 3n+1 for n some non-negative integer . Methodology : proving by contradiction and various properties related to the congruent in order to find the Diophantine equation’s solutions. Main Results : the Diophantine equation tx + (t + 3k)y = z2 has no any solution. Conclusions : the Diophantine equation tx + (t + 3k)y = z2 where t is a positive integer, which is in the form 3n+1 for some non-negative integer n , has no any solution for k, x, y and z are non-negative integers.