Does there exist a positive integer which is a power of 2 such that we can obtain another power of 2 by rearra

Question

Does there exist a positive integer which is a power of 2 such that we can obtain another power of 2 by rearranging it's digit ?

Answer 1

no, it's not possible

Answer 2

there are many such numbers such as 10^2 = 100 rearranging it would give 001 = 1^2

also 11^2 = 121 = 11^2
12^2 = 144 ; 441 = 21^2
13^2 = 169 ; 961 = 31^2;
196 = 14^2

i dont want to give any more here is a list of square

Answer 3

2^n = a
2^m = b

where n and m are both positive integers and by rearranging the digits in a, you end up with b.

It doesn't work up to 2^53, and as the numbers increase it becomes increasingly unlikely. I can't prove it, but it seems very unlikely.