Talk:Powers of 2 not containing Digit Power of 2

Here is the relevant part in 1990: JRM Vol. 22, No. 1, p.76, Solutions to Problems and Conjectures:

*1693. Powers of 2 by Ahmer Yasar Özban, Ankara, Turkey (JRM, 21:1, p.68) <= published 1989


 * Prove or disprove that the only power of $2$ in which no digit is a power of $2$ os $2^{16} = 65536$.

Research by Various Readers


 * Research by various readers certainly indicates that $2^{16}$ may indeed be the only power of $2$ having no digit which is power of $2$. Friend H. Kierstead, Jr. verified the result up to $2^{167}$. Henry Ibstedt showed that if the powers of $2$ contain between $500$ and $10000$ 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.

The rest of the article concerns the obvious generalization by Michael Keith, finding 8 more solutions for powers of 3, 4, 7, 8.

The asterisk next to 1693 probably indicates that this is unsolved. --RandomUndergrad (talk) 03:41, 18 July 2020 (UTC)