Intermezzo

For all integer numbers n bigger than one, the factorial n!, defined to be the result of the multiple product n*(n-1)*(n-2)*...*2*1, cannot be a square number, as was most neatly proven by Erdős. However, the difference between n! and the first square number k^2 that is bigger than n!, is amazingly often a square number:

4! + 1^2 = 5^2
5! + 1^2 = 11^2
6! + 3^2 = 27^2
7! + 1^2 = 71^2
8! + 9^2 = 201^2
9! + 27^2 = 603^2
10! + 15^2 = 1905^2
11! + 18^2 = 6318^2
[ (12-1)! + (19-1)^2 = (89*71-1)^2 ]
13! + 288^2 = 78912^2
14! + 420^2 = 295260^2
15! + 464^2 = 1143536^2
16! + 1856^2 = 4574144^2

I postulate that there are no more such identities. In other words, I postulate that

(ceiling(sqrt(n!)))^2 - n!

cannot be a square number if n is greater than 16.

P.S. It was brought to my attention that the above observation is known and that I am not the first to state the postulate. I was not aware of this.

עב