The problem is to find all positive integers $a$ and $b$ such that $a^2(2^a-a^3)+1=7^b$.
I found a=10, and my intuition tells me there are no more solutions. I've also shown that $a=42k+10$ for some nonnegative integer $k$, but I can't prove anymore than this. (It could help to know that it's from the problems section of a book, so it should have a fairly nice solution.)
Thanks for your help!