Find the values of n leq 7 for which n 1 is a perfect squar

Find the values of n leq 7 for which n! + 1 is a perfect square (it is unknown whether n! + 1 is a square for any n > 7). True or false? For positive integers m and n, (mn)! = m!n! and (m + n)! = m! + n!.

Solution

A).

n<=7 n! +1 is perfect square

so for n=1 1+1 = 2 which is not a sqaure

n=2 2!+1 = 5 which is not a perfect sqaure

n=3 3! +1 = 10 not a sqare

n= 4! = 24+1 = 25 perfect sqare

n=5 120+1 = 121 perfect

n=6 720+1 = not perfect

n=7 5040 +1 =71

so at n= 4,5 7 n!+1 are perfect squares
B).

(mn)! = m!n!

let us take m=3 and n=2

(3*2) ! = 6! = 720

and m!n! = 3! *2! = 6*2 =12

so not true
(m+n)! = m!+n!

(3+2) ! = 5! = 120

where as 3! +2! = 6+2=8

False