The Shuffle Variant of a Diophantine equation of Miyazaki and Togb\'{e}

TL;DR

This paper investigates a variant of a known Diophantine equation, proving unique solutions for the modified equations and extending the results to a related equation involving odd integers.

Contribution

It introduces and analyzes a shuffle variant of a Miyazaki-Togbé Diophantine equation, providing the first solutions and establishing their uniqueness.

Findings

The equation (2am+1)^x + (2m)^y = (2am-1)^z has only two solutions.

The equation b^x + 2^y = (b-2)^z has only two solutions for odd b.

The results extend the understanding of solutions to related Diophantine equations.

Abstract

In 2012, T. Miyazaki and A. Togb\'{e} gave all of the solutions of the Diophantine equations $(2am-1)^x+(2m)^y=(2am+1)^z$ and $b^x+2^y=(b+2)^z$ in positive integers $x,y,z,$ $a>1$ and $b\ge 5$ odd. In this paper, we propose a similar problem (which we call the shuffle variant of a Diophantine equation of Miyazaki and Togb\'{e}). Here we first prove that the Diophantine equation $(2am+1)^x+(2m)^y=(2am-1)^z$ has only the solutions $(a, m, x, y, z)=(2, 1, 2, 1, 3)$ and $(2,1,1,2,2)$ in positive integers $a>1,m,x,y,z$. Then using this result, we show that the Diophantine equation $b^x+2^y=(b-2)^z$ has only the solutions $(b,x, y, z)=(5, 2, 1, 3)$ and $(5,1,2,2)$ in positive integers $x,y,z$ and $b$ odd.