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This is a Quantitative Study study.

October 20, 2025

The Diophantine Equations 6 x + 4y = z2 and 24x + 4y = z2

Key Points

The diophantine equation 6x + 4y = z² has a unique solution (2, 3, 10), indicating a significant finding in number theory.
The equation 24x + 4y = z² presents two solutions: (1, 0, 5) and (2, 5, 40), demonstrating diverse outcomes.
Both equations were analyzed through elementary number theory and Mihăilescu’s theorem, highlighting their importance.
The results emphasize the complexity and uniqueness of solutions in diophantine equations with non-negative integers.

Abstract

This research aims to study and find all solutions of two Diophantine equations 6 x + 4y = z2 and 24x + 4y = z2 where x, y and z are non-negative integers, by using elementary concepts of number theory and Mihăilescu’s theorem. The research results found that the Diophantine equation 6 x + 4y = z2 has the unique non-negative integer solution (x, y, z) = (2,3,10). The Diophantine equation 24x + 4y = z2 has exactly two non-negative integer solutions (x, y, z), which are (1, 0, 5) and (2, 5, 40).

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