## Missouri Journal of Mathematical Sciences

- Missouri J. Math. Sci.
- Volume 29, Issue 2 (2017), 115-124.

### 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 $y$.

#### Article information

**Source**

Missouri J. Math. Sci., Volume 29, Issue 2 (2017), 115-124.

**Dates**

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.mjms/1513306825

**Digital Object Identifier**

doi:10.35834/mjms/1513306825

**Mathematical Reviews number (MathSciNet)**

MR3737291

**Zentralblatt MATH identifier**

06905059

**Subjects**

Primary: 11D41: Higher degree equations; Fermat's equation

Secondary: 26D15: Inequalities for sums, series and integrals 70-08: Computational methods

**Keywords**

$k$-Mass-System of $n$ points (positive integers) Beal's Conjecture Diophantine equation center of mass

#### Citation

Rahimi, Amir M. An Elementary Approach to the Diophantine Equation $ax^m + by^n = z^r$ Using Center of Mass. Missouri J. Math. Sci. 29 (2017), no. 2, 115--124. doi:10.35834/mjms/1513306825. https://projecteuclid.org/euclid.mjms/1513306825