Turkish Journal of Mathematics
Abstract
Let $m$ be a positive integer. We show that the exponential Diophantine equation $ (18m^2+1)^x+(7m^2-1)^y=(5m)^z $ has only the positive integer solution $(x,y,z)=(1,1,2)$ except for $m \equiv 23,47,63, 87 \pmod {120}$. For $m\not\equiv 0 \pmod5$ we use some elementary methods and linear forms in two logarithms. For $m \equiv 0 \pmod 5$ we apply a result for linear forms in $p$-adic logarithms.
DOI
10.3906/mat-1801-76
Keywords
Exponential Diophantine equations, linear forms in the logarithms
First Page
1990
Last Page
1999
Recommended Citation
ALAN, M (2018). On the exponential Diophantine equation $(18m^2+1)^x+(7m^2-1)^y=(5m)^z$. Turkish Journal of Mathematics 42 (4): 1990-1999. https://doi.org/10.3906/mat-1801-76
