Powers of 2 not containing Digit Power of 2

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Theorem

$2^{16} = 65 \, 536$ is the only known power of $2$, up to $2^{31 \, 000}$, whose digits do not contain $1$, $2$, $4$ or $8$.


Proof

This has been demonstrated by an exhaustive search.

$\blacksquare$


Historical Note

This question appears to have first been raised by Ahmer Yasar Özban in $1989$.

While no formal attempt has been made to solve it, several searches were made in response, as follows:

Friend H. Kierstead, Jr. verified the result up to $2^{167}$.
Henry Ibstedt showed that if the powers of $2$ contain between $500$ and $1000$ digits, the digits $1, 2, 4, 8$ occur fairly normally.
Douglas J. Lanska checked the powers of $2$ up to $2^{3320}$, finding no other solution.
L.M. Leeds searched through $2^{20703}$, also finding no other solution.
Finally, Charles Ashbacher went to $2^{31000}$, which contains $9332$ digits, with the same result.


Sources