solving this equation using prime numbers

Question

Solve in $\mathbb{Z}$ the following equation:

$3^x$+$3^y$=$738$,

using prime numbers concept and decomposition in prime factors...

I noticed that the above equation is symmetrical to $x$ and $y$, so we can assume $x$ $\leq$ $y$ ... but I can not find a way to solve it. Also, clearly: $738$=$3^2$$*$$2$$*$$41$.

Thanks for your time.

Answer 1

Note that $3^x+3^y=3^x\cdot(1+3^{y-x})$, so we need $3^x=3^2$and $3^{y-x}=2\cdot 41-1$. (You will note that the complete factorization of $738$ was not needed, only the number of $3$'s).

Answer 2

Nice answer from Hagen von Eitzen, here is a different method: in base $3$ we have $$738=[1000100]_3\ ,$$ so the powers are $3^6$ and $3^2$.

Admittedly this does not really use prime factorisation so perhaps it is not actually an answer to the question as posed.