February 28, 2024Open Access

On the Exponential Diophantine Equation 7ˣ − 5ʸ = z²

Key Points

The only solution to the equation 7^x - 5^y = z^2 is (0,0,0), demonstrating uniqueness.
Key findings arise from examining congruence theorems and Catalan's conjecture in the context of the equation.
In-depth analysis involves computational methods combined with fundamental concepts from number theory for validation of findings.. outcomes enhance understanding of Diophantine equations in mathematical theory.

Abstract

In this paper, we address the exponential Diophantine equation 7x −5y = z2 , seeking non-negative integer solutions for x, y, and z. Usingmany congruence theorems and Catalan’s conjecture, we prove theexistence of a single solution. Our analysis shows that (x,y,z)=(0,0,0)is the only possible solution to the problem. We prove the validity ofthis claim by a thorough analysis of computational methods and conceptsfrom number theory. This outcome advances our knowledge of exponentialDiophantine equations and sheds light on how prime numbers andexponentiation interact in these kinds of mathematical investigations.

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Cite This Study

Budee U Zaman (Wed,) studied this question.

synapsesocial.com/papers/68e7741eb6db6435876e8f69 https://doi.org/https://doi.org/10.31219/osf.io/ycn4u

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