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I was wondering if for almost all exponents $x$, $2^x$ written in decimal would contain at least one 0 digit.

I've checked with a computer up to $ 10^5 $ and I didn't find any exponent which doesn't contain a 0 greater than 86.

Is that correct generally? How can you prove/disprove it?

I intuitively think it's correct, because as a number has more base 10 digits I would expect it to be more rare for none of its digit to be 0.

Command Master
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    I think your intuition is probably correct, but this is one of those questions that turns out to be extremely hard to answer—I believe this is an open problem. – Greg Martin Apr 23 '21 at 05:21
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    Worth mentioning this question which also is about the decimal expansion of powers of two. Both are about the same in difficulty, i. e. very challenging, I'd say. – Kyan Cheung Apr 23 '21 at 05:24
  • Also related: https://math.stackexchange.com/questions/1199682/is-there-a-k-such-that-2n-has-6-as-one-of-its-digits-for-all-n-ge-k – leonbloy May 11 '21 at 22:17
  • Note that the question in the title is the opposite of what you actually ask. Almost surely, there is no more power of $2$ conataining no $0$ in the decimal expansion, but I doubt we can prove this. – Peter Mar 11 '22 at 08:48

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