On an exponential diophantine equation

Question

I am trying to find all integer solutions of $5^x + 12 ^y$ = $13^z$. The obvious (and pursued) solution is $(2, 2, 2)$, and no others. I've tried to use an appropriate modular arithmetic, but to no avail. Any ideas on how to solve this?

Answer 1

For this special case the problem is easier and indeed has been solved. Let $(a,b,c)$ be a Pythagorean triple, i.e., integers satisfying $a^2+b^2=c^2$. Then it is conjectured that the Diophantine equation $$ a^x+b^y=c^z $$ has the only positive integral solution $(x,y,z)=(2,2,2)$. For $(a,b,c)=(3,4,5)$ this has been shown by Sierpinski in $1956$, and for many other triples the result is also known, see for example the article "The Diophantine Equation $a^x+b^y=c^z$ by N. Terai of $1994$. For $(a,b,c)=(5,12,13)$ the result has been proved by R. Scott in $1993$. The result is also contained in a theorem of Maohua Le, "A Note on Jesmanowicz Conjecture" (1995):

Theorem (Le): Let $a=r^2-s^2$, $b=2rs$, $c=r^2+s^2$ with $2\mid rs$, $4\nmid rs$, $c=p^n$ for some odd prime $p$, then $a^x+b^y=c^z$ has only the solution $(x,y,z)=(2,2,2)$.

Fpr $r=3$ and $s=2$ we obtain the case $(a,b,c)=(5,12,13)$.