Natural answers of $(n-1)(n)(n+1)+1=x^2$

Question

I know that the multiplication of four consecutive numbers plus 1 is always a square. This means that for $n,x\in\mathbb{N}$: $$(n)(n+1)(n+2)(n+3)+1=x^2$$

After this, I thought about the multiplication of three consecutive numbers plus 1, so for $n,x\in\mathbb{N}$: $$(n)(n+1)(n+2)+1=n^3+3n^2+2n+1=x^2$$ or, for $n>1\land n, x\in\mathbb{N}$ (so that an elliptic curve is obtained instead): $$(n-1)(n)(n+1)+1=n^3-n+1=x^2$$

I have found two solutions up to $2642245, 2642246, 2642247$ that were obtained with a C++ script. These are: $$2\cdot3\cdot4+1=25=5^2\\4\cdot5\cdot6+1=121={11}^2\\55\cdot56\cdot57+1=175561={419}^2$$

Are there any more solutions? Is there a finite amount of them? Thanks in advance.

Answer 1

jagy@phobeusjunior:~$ date
    Mon Mar 26 15:31:40 PDT 2018
    jagy@phobeusjunior:~$ sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath Version 6.9, Release Date: 2015-10-10                     │
│ Type "notebook()" for the browser-based notebook interface.        │
│ Type "help()" for help.                                            │
└────────────────────────────────────────────────────────────────────┘
sage:  E = EllipticCurve([0,0,0,-1,1])
sage: E
Elliptic Curve defined by y^2 = x^3 - x + 1 over Rational Field
sage:   E.integral_points()
[(-1 : 1 : 1),
 (0 : 1 : 1),
 (1 : 1 : 1),
 (3 : 5 : 1),
 (5 : 11 : 1),
 (56 : 419 : 1)]
sage: quit
Exiting Sage (CPU time 0m1.32s, Wall time 0m48.19s).
jagy@phobeusjunior:~$ date
Mon Mar 26 15:32:57 PDT 2018
jagy@phobeusjunior:~$ 

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