An interesting observation about factorials

A factorial, is a mathematical tool, usually used
in probability. If you have, say 5! (! is the sign),
It equals 5*4*3*2*1. So the factorial of any
number is represented by the product of all the
numbers below it.

From 1 to 10 it's results are: -

1! = 1 , 2! = 2 , 3! = 6 , 4! = 24 , 5! = 120 , 6! = 720 , 7! = 5040 ,
8! = 40320 , 9! = 362880 , 10! = 3628800

There is something interesting about factorials that
I found out for numbers above 3.

q! + 1 =  p^2 (p represents primes)

 4! + 1 = 25 = 5^2

5! + 1  = 121 = 11^2

6! + 1 = 721 = 29^2

7! + 1 = 5041 = 71^2

8! + 1 = 40321 = (200. 8208..... ^ 2) So, the pattern breaks at 8, but what about higher factorials?

9! + 1 = (602. 39.... ^ 2) Not with nine

10! + 1 = 3628801 = (1904. 94..... ^ 2) Not with ten either

It seems this rule only works for numbers below 8 and above 3.