Does the Solution to the Non-linear Diophantine Equation 3^x+35^y=Z² Exist?

Abstract

This paper investigates the solutions (if any) of the Diophantine equation 3^x + 35^y = Z², where  , x, y, and z are whole numbers. Diophantine equations are drawing the attention of researchers in diversified fields over the years. These are equations that have more unknowns than a number of equations. Diophantine equations are found in cryptography, chemistry, trigonometry, astronomy, and abstract algebra. The absence of any generalized method by which each Diophantine equation can be solved is a challenge for researchers. In the present communication, it is found with the help of congruence theory and Catalan’s conjecture that the Diophantine equation 3^x + 35^y = Z² has only two solutions of  (x, y, z) as  (1, 0, 2) and (0, 1, 6) in non-negative integers.