The Shuffle Variant of a Diophantine equation of Miyazaki and Togbé

Abstract

In 2012, T. Miyazaki and A. Togbé 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 Togbé). 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.

12 pages, accepted for publication