In IMO Shortlist 2013, there is a number theory problem:

Determine whether there exists an infinite sequence of nonzero digits $a_1,a_2,a_3,...$ and a positive integer $N$ such that for every integer $k>N$, the number $\overline{a_ka_{k-1}...a_1}=\sum_{i=1}^ka_i10^{i-1}$ is a perfect square.

This is a very interesting problem and the generalizations of this problem can have research value. So I guess that this problem is based on some research paper.

Is there any paper about or relates to this problem?

  • 3
    $\begingroup$ I assume the notation $\overline{a_ka_{k-1}...a_1}$ means the number $\sum_{i=1}^k a_i10^{i-1}$. $\endgroup$
    – YCor
    Sep 28, 2022 at 6:43
  • 2
    $\begingroup$ @YCor I would assume that it means the repeating decimal, i.e. $(1-10^{-k})^{-1}\sum_{i=1}^ka_i10^{1-i}$, and that the OP wants to be a square in $\mathbb{Q}$. But it would be best for the OP to clarify. $\endgroup$ Sep 28, 2022 at 9:15
  • 8
    $\begingroup$ @NeilStrickland I think YCor's interpretation is correct. The overline notation is pretty standard in olympiad circles to denote the number represented by a decimal string. $\endgroup$
    – Wojowu
    Sep 28, 2022 at 10:40

1 Answer 1


This problem and its solution appeared in a 1981 issue of Crux Mathematicorum, volume 7, issue 9, pages 280-282.

There it was proven by L. Csirmaz that the longest possible sequence is $25, 625, 5625, 75625,275625$.

The problem was first posed by K.S. Williams in 1980. I copy below the results from an initial computer search, which was then shown to be exhaustive in 1981.


A variety of proofs can be found at AoPS.


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