## Missouri Journal of Mathematical Sciences

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

Amir M. Rahimi

#### 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

#### 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

#### References

• I. G. Bashmakova, Diophantus and Diophantine Equations, MAA, Washington, D. C., 1997.
• M. L. Boas, Mathematical Methods in the Physical Sciences, John Wiley & Sons, Inc., New York, NY, 1966.
• D. E. Flath, Introduction to Number Theory, John Wiley & Sons, Inc., New York, NY, 1989.
• R. D. Mauldin, A generalization of Fermat's Last Theorem: The Beal conjecture and prize problem, Notices of the AMS, 44.11 (1997), 1436–1439.
• A. M. Rahimi, A physical approach to Goldbach's Conjecture and Fermat's Last Theorem, Libertas Mathematica, 28 (2008), 149–153.
• K. H. Rosen, Elementary Number Theory and Its Applications, 3rd ed., Addison Wesley Longman, Inc., Texas, 1992.
• G. B. Thomas, Calculus, Addison-Wesley Inc., Kentucky, 2005.
• A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. Math., 141 (1995), 443–551.
• Wikipedia, Beal Conjecture, \tthttp://en.wikipedia.org/wiki/Beal_conjecture.