Are there infinitely many squares in the set $$\{\sum\_{j=1}^m j^2: m\in\mathbb{N}\} ?$$



Squares in the set $\{\sum_{j=1}^m j^2: m\in\mathbb{N}\}$
We have $\sum\_{j=1}^m j^2 = \frac{m(m+1)(2m+1)}6$. Hence, the question reduces to finding integral points on the elliptic curve:
$$y^2 = \frac{x(x+1)(2x+1)}6.$$
Turning it to Weierstrass equation with integer coefficients, we get:
$$(72y)^2 = (12x)^3 + 18 (12x)^2 + 72 (12x).$$


For many curves, integral points can be readily found by SageMath, and in this case we are lucky:



```
sage: EllipticCurve([0,18,0,72,0]).integral_points()
[(-12 : 0 : 1), (-9 : 9 : 1), (-8 : 8 : 1), (-6 : 0 : 1), (0 : 0 : 1), (6 : 36 : 1), (12 : 72 : 1), (288 : 5040 : 1)]

```

So, the only solutions are $m=\frac{0}{12}=0$, $m=\frac{12}{12}=1$, and $m=\frac{288}{12}=24$.


