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)


 * Thank you, that information has been incorporated.
 * Is it possible to check whether the article "Research by Various Readers" is actually written by Ashbacher, or whether it is actually the work of Madachy in the article titled "Solutions to Problems and Conjectures"? --prime mover (talk) 07:40, 18 July 2020 (UTC)