Elif Kizildere, Gökhan Soydan, Qing Han, Pingzhi Yuan: The shuffle variant of a Diophantine equation of Miyazaki and Togbé, 243-254

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.

Key Words: Exponential Diophantine equation, Baker's method.

2010 Mathematics Subject Classification: Primary 11D61; Secondary 11J86.