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Following up on this question,

https://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102

is anything known about the sequence of those repdigits (i.e. $1111, 44444$), which are expressible as the sum of squares of consecutive numbers?

In particular, are there infinitely many of them?

Edit: The question is not restricted to base $10$, but this is the case I am most interested in. A heuristic argument (see above link) suggests that there may be only finitely many after all. For all I know so far, this may or may not depend on the base.

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  • $\begingroup$ The first few are 0, 1, 4, 5, 9, 55, 77, 1111, 44444, 444444. $\endgroup$
    – Gerry Myerson
    Commented May 16, 2016 at 22:58
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    $\begingroup$ Those are repdigits, not one-digit numbers. One-digit numbers range from $1$ to $9$. en.wikipedia.org/wiki/Repdigit $\endgroup$
    – Douglas Zare
    Commented May 17, 2016 at 0:44
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    $\begingroup$ @DouglasZare For a moment I was really excited about being able to solve a number theory problem on MO . . . $\endgroup$
    – Noah Schweber
    Commented May 17, 2016 at 2:13
  • $\begingroup$ Is the question restricted to decimal numbers and, can the answer regarding finiteness depend on the basis? $\endgroup$
    – Manfred Weis
    Commented May 17, 2016 at 2:59

1 Answer 1

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Here is partial answer which might give infinitely many cheap solutions in some base. If I am not mistaken, there are no cheap solutions in bases up to $100$.

Let $f_n(x)$ be the sum of $n$ consecutive squares, there is closed form form for it.

Base $B$ repdigit $ddd\ldots$ is of the form $d\frac{B^y -1}{B-1}$.

After some rewriting, solution is $(B-1)f_n(x)+d=dB^y$.

Cheap solutions are when the LHS is of the form $c(ax+b)^2$ for integers $a,b,c$ and there are no congruence obstructions for the RHS.

Sufficient condition for this is the discriminant $D$ of the LHS to vanish.

$D$ is polynomial in $n,d,B$ and we are looking for solutions with some restrictions. For fixed $d,B$, it is univariate in $n$ and positive root is solution.

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