Algebraically proving a number is a square?

Question

Hey everyone, I was wondering if anyone knew if it was possible to algebraically prove that a number is a square number? For example, for some strange reason X(X+1)(X+2)(X+3)(X+4) + 1 = a square number... but is there any way to show this algebraically? I can show it through numerous examples, but that isn't a proof.

Thanks!

Oh I see I screwed up! I added an extra X+y, it should be X(X+1)(X+2)(X+3) + 1 = a square number, which works! I got it now, thanks a lot for helping!

Answer 1

I tried... but realized it wasn't true...

I should have verified your assumptions. Certainly with x=2 this doesn't work.

Answer 2

Are you sure about that equation being true for all x?

While it's true that x = 1 gives you 121, which is 11 squared, I don't see that for x = 2. That gives you 721, which is not a square.

I don't think that does have a proof.

Edit: Ah, your additional detail makes sense. Thank you for clarifying. With that in mind, I have to go with the person below me.

Further edit: Whoa, akilesh_12003, you can't just call that a proof. To prove a statement, you have to show that it's true for all instances. Just saying that it's true for 2 doesn't prove a thing. By that reasoning, the original formula of X(X+1)(X+2)(X+3)(X+4) + 1 = n^2 is true all because x=1 is true.

You don't prove a whole statement by showing one example. It is not logical to say, "All red things are fire trucks. As proof, I give you a fire truck."

Answer 3

The additional detail makes a lot of sense. Let me prove it true taking X = 2 as an example
2(2+1)(2+2)(2+3) + 1 = 2(3)(4)(5) +1
= (6)(20) + 1
= 120 + 1
= 121 which is a square

Answer 4

Your formula is not true for all X
say if X=8
then
(8)(9)(10)(11)(12)+1=95041 which is NOT a square number

maybe you made a mistake somewhere...

Given the correction you make then its easy:
X(X+1)(X+2)(X+3)+1=
X^4+6X^3+11X^2+6X+1=
( X^2+3X+1) ^2
so yes, it is always a square