New problem.
Does there exist any two distinct integers a and b such that 2^a is just a rearrangement of the digits of 2^b?
A rearrangement of the digits is when the same digits are re-ordered to make a new number: 12345 is a rearrangement of 45321. It is not a rearrangement of 4532 or 453212 or 4532.1 for example.
If yes, provide an example. If no, show why not.
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