Which nontrivial powers can be obtained by rearrangement of digits of powers of 2 ?
source - own curiosity, as usual in such cases...
Here are 3 interesting cases I discovered:
powers of 3 -- obviously impossible (sum of digits).
powers of 4 -- trivial.
powers of 5 -- possible: $2^9=512\rightsquigarrow 125=5^3$.
powers of 6 -- obviously impossible.
powers of 7 -- possible: $2^{10}=1024\rightsquigarrow 2401=7^4$.
powers of 8 -- trivial.
powers of 9 -- obviously impossible.
powers of 10 -- obviously impossible.
powers of 11 -- ??
powers of 12 -- obviously impossible.
powers of 13 -- ??
powers of 14 -- possible: $2^{14}=16384\rightsquigarrow 38416=14^4$.
powers of 15 -- obviously impossible.
powers of 16 -- trivial.