TITLE: $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$
QUESTION [5 upvotes]: How do I show that the diophantine equation $2^x+7^y=19^z$ has no solution in positive integers $x$, $y$, $z$

REPLY [9 votes]: Going modulo $6$, the equation gives $$2^x+1\equiv 1 \pmod 6$$So, $$2^x\equiv 0 \pmod 6$$But no power of $2$ is divisible by $6$.
Which shows that no solutions exist. Hence proved.