Turkish Journal of Mathematics
DOI
10.3906/mat-1801-76
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.
Keywords
Exponential Diophantine equations, linear forms in the logarithms
First Page
1990
Last Page
1999
Recommended Citation
ALAN, MURAT
(2018)
"On the exponential Diophantine equation $(18m^2+1)^x+(7m^2-1)^y=(5m)^z$,"
Turkish Journal of Mathematics: Vol. 42:
No.
4, Article 35.
https://doi.org/10.3906/mat-1801-76
Available at:
https://journals.tubitak.gov.tr/math/vol42/iss4/35
