Does the Solution to the Non-linear Diophantine Equation 3<sup>x</sup>+35<sup>y</sup>=Z<sup>2</sup> Exist?
This paper investigates the solutions (if any) of the Diophantine equation 3x + 35y = 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 3x + 35y = Z2 has only two solutions of (x, y, z) as (1, 0, 2) and (0, 1, 6) in non-negative integers.
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