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
- 1989: Ahmer Yasar Özban: Problems and Conjectures: $\text Q 1693$. Powers of $2$ (J. Recr. Math. Vol. 21, no. 1: p. 68)
- 1990: Solutions to Problems and Conjectures: $\text Q 1693$. Powers of $2$: Research by Various Readers (J. Recr. Math. Vol. 22: p. 76)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $65,536$