An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass

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 $y$.