Solution of the Diophantine Equation 163a + 665b = c2

Abstract

Numerous fields, including algebra, trigonometry, astronomy, chemistry, computer science, and astronomy, use Diophantine equations. These equations are essential for resolving algebraic word problems, chemical reaction balancing difficulties, and problems demonstrating a number's irrationality. The Diophantine equation 163^a + 165^b = c², where a, b, c are non-negative integers, was examined by the authors of this manuscript. They demonstrated that there is no solution to this Diophantine equation in the set of non negative integers.