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So, for people who are stuck:
This problem is very confusingly formulated. What it wants from you is, given a string, find all substrings that represent powers of 2.
There are powers up to 800, that's a 800-bit integer, you ain't going to find one in any language. Hint: use strings.
Then, you're going to need to find lots of substrings in a string. Others mentioned Aho-Corasick. I managed to do it just with tries. Aho-Corasick is essentially tries plus some optimization. So same thing, but simpler.
this question is word soup.
Don't forget the 2^0 :)
Also some hints for people who are stuck: