# Status of a conjecture about powers of 2

I recently saw a conjecture on a blog ( http://blog.tanyakhovanova.com/?p=311 ) which the author refers to as the 86 conjecture. The conjecture claims that all powers of 2 greater than $2^{86}$ have a zero in their base 10 representation.

At first glance this seems like a numerical curiosity, and I wasn't sure that this was an interesting or deep mathematical question. I wanted to ask the community if they knew of any existing work that deals with either this type of question or something similar.

I tried searching for 86 conjecture on Google and didn't get anything useful.

• Decimal expansions of powers are pretty chaotic creatures. I spent a lot of time trying to bound the total number of zeros in powers of five. I couldn't even linearly bound them. But it is possible to show that any amount of zeros in a row eventually appears in the powers of five (section 10: arxiv.org/abs/1910.13829). For numbers coprime to $10$, showing any amount of zeros in a row appear in its powers is easy. Just note that $n$ is a unit in $(\mathbb{Z}/10^k\mathbb{Z})^\times$. – Onno Cain Nov 12 at 19:26

Richard Guy's Unsolved Problems in Number Theory, Problem F24, mentions only that Dan Hoey has verified this conjecture for $2^n$ up to $n = 2,500,000,000$. Guy tends to be fairly complete in his references. Since he doesn't give any others I doubt there's much more out there.