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Prove that there does not exist any positive integer P which is a power of 2 with the proviso that the digits of P (in the base ten representation) can be permuted to form a different power of 2.

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The difference of two base 10 numbers which are digit permutations of one another is always a multiple of 9.

Note that, for a difference of two powers of 2 to be a multiple of 9, the powers must differ by a multiple

of 6. This is easy to see if you think of the two numbers in base 2 -- 9 is 1001 and the difference of the

powers of 2 is a sequence of 1s surrounded by 0s. But, powers of 2 differing by a multiple of 64 (26)

makes them differ in the number of decimal digits they contain. Therefore, they cannot be permutations of each other.

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