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

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.

• Thanks! The weird thing is that it was well written both in the "source code" and in the preview. I have had to cut and paste the line. – Manuel B Mar 26 '18 at 22:04
• you missed $5^3 - 5 + 1 = 11^2$ – Will Jagy Mar 26 '18 at 22:34
• @WillJagy Yes, you're right. The best part of it is that it seems to be a problem of the C++ compiler I am using, not of the code I have written. – Manuel B Mar 26 '18 at 23:24
• Notice that $5$, $11$ and $419$ are all prime numbers.... – Feeds Mar 28 '18 at 10:50

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:~$date Mon Mar 26 15:32:57 PDT 2018 jagy@phobeusjunior:~$


.................

• Do you know/have any way to know if those are the only results, or if SageMath stops at a certain point after checking for all points belonging to the curve? – Manuel B Mar 26 '18 at 22:50
• @ManuelB these are all. If sage accepts the problem, then it finishes it. There are fairly small bounds on the numbers sage will accept as coefficients, something like absolute value up to 10,000. Perhaps you should look up sage online and read the documentation. Also, downloading is free. – Will Jagy Mar 26 '18 at 22:57