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

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$(2,1,1,2,2)$inpositiveintegers$a>1,m,x,y,z$. 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.