factorials to perfect squares

[Tahmid Hasan]

Find all positive integers $n$ for which $(n + 2)! + (n - 2)!$ is a perfect square.

[Masum]

Find all positive integers $n$ for which $(n + 2)! + (n - 2)!$ is a perfect square.

Rewrite as $(n-2)!((n+2)(n+1)n(n-1)+1)$
But $(n+2)(n+1)n(n-1)+1=(n^2-n+1)^2$
So we need $(n-2)!$ a square.
Theorem (ERDOS):
The product of more than two consecutive natural numbers is not a perfect power.
Corollary:
$a!$ is never perfect square for $a>1$
So we must have $n-2=0,1\Rightarrow n=2,3$

[Tahmid Hasan]

Theorem (ERDOS):
The product of more than two consecutive numbers is not a perfect power.

could you please give the proof or a link to the proof.

[Masum]

I hardly have that courage, probably you didn't understand who Erdos is! I have a pdf on its proof written by Erdos and Seilburg. But how dare I upload it?

[Tahmid Hasan]

probably you didn't understand who Erdos is!

i do know erdos and i know the erdos-mordell inequality.
however,is the proof copyrighted?

[Mohaimin]

Theorem (ERDOS):
The product of more than two consecutive numbers is not a perfect power.

I believe it should be:
The product of more than two consecutive positive integers is not a perfect power.

You should be more careful when stating a theorem.

[Masum]

Yes, so edited.

[Masum]

probably you didn't understand who Erdos is!

i do know erdos and i know the erdos-mordell inequality.
however,is the proof copyrighted?

I don't understand what is the connection between knowing about Erdos and knowing Erdos and Mordel inequality. Whatever, now I can't upload it. But just search in google with keywords Erdos and Seilburg pdf on consecutive integers. You will surely get it. Then read it and you will understand if the proof is copyrighted or else.

[Tahmid Hasan]

I was just a little confused about these theorems.there are many theorems of Erdos and i mixed them up.sorry

[Masum]

I gave you the keywords you need. Why don't you download all the pdfs?