I've the following number:

$$12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag1$$

Now I know that $n\in\mathbb{N}^+$ and $n\ge3$ (and $n$ has a given value) besides that $x\in\mathbb{N}^+$ and $x\ge2$.

I want to check if the number is a perfect square, so I can rewrite $(1)$ as follows:

$$y^2=12\left(n-2\right)^2x^3+36\left(n-2\right)x^2-12\left(n-5\right)\left(n-2\right)x+9\left(n-4\right)^2\tag2$$

Where $y\in\mathbb{Z}$.

In this problem I've: $n=71$, the number is equal to;

$$y^2=57132x^3+2484x^2-54648x+40401\tag3$$

So, I used SageMathCell to look for the integral points on the elliptic curve and the code that was used is the following:

```
E = EllipticCurve([0, β, 0, γ, δ])
P = E.integral_points()
for p in P:
if p[0] % α == 0:
print(p[0] // α, p[1] // α)
```

I found the coeficients I need to use using equation $(2)$ and $(3)$ (but I do not know if they are corect):

- $$\alpha=12(71-2)^2=57132\tag4$$
- $$\beta=36(71-2)=2484\tag5$$
- $$\gamma=-144(71-5)(71-2)^3=-3122149536\tag6$$
- $$\delta=1296(71-4)^2(71-2)^4=131871507195024\tag7$$

So the final code looks like:

```
E = EllipticCurve([0, 2484, 0, -3122149536, 131871507195024])
P = E.integral_points()
for p in P:
if p[0] % 57132 == 0:
print(p[0] // 57132, p[1] // 57132)
```

But I found no solutions and it should give at least one solution at $x=1585$.

What mistake have I made?

`E = EllipticCurve([0, 2484, 0, -3122149536, 131871507195024])`

, the command`P = E.integral_points()`

hasn't finished after more than an hour. $\endgroup$