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November 2017 An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass
Amir M. Rahimi
Missouri J. Math. Sci. 29(2): 115-124 (November 2017). DOI: 10.35834/mjms/1513306825

Abstract

This paper takes an interesting approach to conceptualize some power sum inequalities and uses them to develop limits on possible solutions to some Diophantine equations. In this work, we introduce how to apply center of mass of a $k$-mass-system to discuss a class of Diophantine equations (with fixed positive coefficients) and a class of equations related to Fermat's Last Theorem. By a constructive method, we find a lower bound for all positive integers that are not the solution for these type of equations. Also, we find an upper bound for any possible integral solution for these type of equations. We write an alternative expression of Fermat's Last Theorem for positive integers in terms of the product of the centers of masses of the systems of two fixed points (positive integers) with different masses. Finally, by assuming the validity of Beal's conjecture, we find an upper bound for any common divisor of $x$, $y$, and $z$ in the expression $ax^m+by^n = z^r$ in terms of $a, b, m({\rm or} ~n), r$, and the center of mass of the $k$-mass-system of $x$ and $y$.

Citation

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Amir M. Rahimi. "An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass." Missouri J. Math. Sci. 29 (2) 115 - 124, November 2017. https://doi.org/10.35834/mjms/1513306825

Information

Published: November 2017
First available in Project Euclid: 15 December 2017

zbMATH: 06905059
MathSciNet: MR3737291
Digital Object Identifier: 10.35834/mjms/1513306825

Subjects:
Primary: 11D41
Secondary: 26D15, 70-08

Rights: Copyright © 2017 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.29 • No. 2 • November 2017
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