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I don't think a number whose decimal expansion looks like 10010111010110001 can be a square number. However, I just can't prove it.

More precisely, does there exist a positive square integer, whose decimal expansion consists only of $0$s and $1$s, apart from the obvious $100^n$, $n\ge 0$?

And if it exists, can it be a power of 3?

YCor
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mutton
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    0, and $10^{2n}$ are square numbers whose digits are 0s and 1s – Gabriel C. Drummond-Cole Mar 21 '22 at 07:39
  • You have asked n the wrong forum. – Gerald Edgar Mar 21 '22 at 09:24
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    i think in base 2 it is true – Pietro Majer Mar 21 '22 at 10:41
  • Please re-ask on math.stackexchange.com. – Charles Mar 21 '22 at 18:52
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    I've edited the question, which seems reasonable to me here. Indeed heuristics indicate that in base $p$ there should be only finitely many solutions: there are $2^n$ candidates in the $\sim p^n$ numbers between $p^n$ and $p^{n+1}$ to have such an expansion, and there are $p^{n/2}$ squares therein. Assuming $p\ge 5$, the probability that some given $(1,0)$-number being a square is $p^{-n/2}$ so the probability that at least one being a square should be $\le (2/\sqrt{p})^n$ and hence, summing, the probability to have a solution $\ge p^n$ should be in $O((2/\sqrt{p})^n)$ too. – YCor Mar 22 '22 at 08:20
  • Experiments: in base $4$ there seems to have infinitely many solutions. In base $5$ I only found $946337894401=972799^2$, which is $111001100000110101$ in base $5$. In base $7$ I only found $400=20^2$, which is $1111$ in base $7$. In base $8$ I only found $9$ and $136352329=11677^2$, the latter being $1010111111$ in base $8$. (Of course in base $n^2-1$ one always has $n^2$ written $11$.) In bases $6,9,10$ I found no solution. (In base $10$ I think I checked up to $10^{40}$.) – YCor Mar 22 '22 at 08:30
  • I made a typo in my first comment. I mean "heuristics indicate that in base $p\ge 5$ there should be only finitely many solutions". – YCor Mar 22 '22 at 08:33
  • I just found out that this question has already been asked and it's known as Crux problem 909. – mutton Mar 22 '22 at 10:36
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    More precisely, this is a duplicate of https://mathoverflow.net/questions/22/can-n2-have-only-digits-0-and-1-other-than-n-10k – YCor Mar 22 '22 at 10:42
  • This question also needs more focus. You asked two questions on the same post. –  Mar 16 '23 at 11:12

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