Find $(m,n)$ where $\sum\limits_{r=1}^n r = \sum\limits_{r=1}^m r^2$.

Question

I was intrigued to find that the sum of the first $10$ integers is equal to the sum of the first $5$ squares. I then tried looking for other similar cases, i.e. by consider the following problem:

Find pairs of $(m,n)$ where $$\sum_{r=1}^n r = \sum_{r=1}^m r^2$$

Converting the summations to closed-form and rearranging gives $$3n(n+1)=m(m+1)(2m+1)$$ How would you go about solving this?

Using numerical methods and considering values for $m$ from $1$ up to $1000$ I found the following solutions: $$(m,n)=(1,1), (5,10), (6,13), (85,645)$$

Could there be more? Or are these the only ones? What other methods would you suggest to approach this problem?

Answer 1

A modern approach to this kind of problems:

From $3n(n + 1) = m(m+1)(2m + 1)$ we get: $$y^2 + 18y = x^3 + 9x^2 + 18x,$$ where $y = 18n$ and $x = 6m$.

It suffices to find integral points on this curve. This is easily done with a computer algebra system.

Paste the following code into this page and press "Evaluate".

E = EllipticCurve([0, 9, 18, 18, 0])
for (x, y, z) in E.integral_points():
    if x % 6 == 0:
        print(x // 6, y // 18)

We get the output:

(-1, 0)
(0, 0)
(1, 1)
(5, 10)
(6, 13)
(85, 645)

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It would seem a bit unfair that I simply post a code without any explanation on how it works.

Since I don't want to jump into the details of elliptic curves, I will talk briefly from an algorithmic point of view.

The computation of integral points on an elliptic curve (over $\mathbb Q$, with a model over $\mathbb Z$) has two steps. The first step is to find a basis of the Mordell-Weil group, i.e. the group of $\mathbb Q$-rational points on the curve; and the second step is to find which linear combinations of these $\mathbb Q$-rational points give rise to integral points.

For the first step, there is no known algorithm which can do it for any elliptic curve over $\mathbb Q$. However, powerful methods (mainly the descent methods + Heegner point method) have been developped, so that in "most cases" we can do this step with this or that method. These, I believe, were developped quite early, probably in the 1920's (except the Heegner point method).

For the second step, the key point is Baker's explicit bound on the integral points, known as Baker's theorem. This was developped in the late 1960's.

However, while Baker's bounds are effective, they are by no means efficient, meaning that they are usually too large to be useful. The solution is the LLL algorithm, which will dramatically reduce the bounds of Baker, eventually to a practically small value. This algorithm is not available until the 1980's.

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Therefore we see the value of the paper by Frinkelstein: it was published in 1972, when the last ingredient LLL was still unavailable...

It's amazing how science and technology changed our lives, and our mathematical researches...