Find all the possible solutions of $a^b+b^a=17$ where $a, b$ are natural numbers.
I get $a$ can't be equal to $b$ as if they are then they won't be natural numbers.
Without loss of generality, we can say that $b \gt a$.
How should I proceed. I am not able to find such numbers but how should I do it?
Should I write $a^b=17-b^a$ and analyse all the possibilities?
Or should I apply log both sides?
I can't proceed with anything.
Please help.
Well, I mean, just check every possible pair, the number of cases is not very big. The possible pairs are $(16,1)$ and $(2,3)$, you see that by hand. If you take $a$ equal to 1,2 or 3 you get these 2, and for $a$ greater than 4 you get no solutions except for the case $a=16$ which is already covered