<s>[INST] Proof that $n!$ is divisible by $(n+1)^2$ for $n=xy+x+y$ I noticed the other day that for $n>8, n\in\mathbb{N}$, the factorial $n!$ seems to be divisible by $(n+1)^2$ when $n$ can be written in the form $xy+x+y$ (where $x,y\geq1$ and $\in\mathbb{N}$). Some examples:


*

*$n=14=2 \times 4+2+4$ (i.e. $x=2$, $y=4$), and we have $14!|15^2$

*$n=15=3 \times 3+3+3$ (i.e. $x=y=3$), and we have $15!|16^2$,

*$n=19=3 \times 4+3+4$ (i.e. $x=3$, $y=4$), and we have $19!|20^2$,

*...


This seems to hold for the first $15$ values I checked, so it seemed natural to try to prove it for all $n=xy+x+y$, however I could not find a proof. Here is my attempt:
Plugging $xy+x+y$ into the expressions above and noticing that $xy+x+y+1=(x+1)(y+1)$, we need to prove that
$$
(xy+x+y)!|(x+1)^2(y+1)^2,
$$
or
$$
\Gamma((x+1)(y+1))|(x+1)^2(y+1)^2.
$$
It is easy to prove that 
$$
\Gamma((x+1)(y+1))|(x+1)(y+1)
$$
however I am struggling to prove the same for $(x+1)^2(y+1)^2$. Any hints?
 [/INST] $\color{Green}{\text{Question}}$ : 
Find all integers $n$; 
such that $\color{Green}{(n+1)! \mid n!} $ . 


$\color{Blue}{\star \ \ \ \ \text{condition}}$ : 
Now suppose that $n+1$ can be written 
in the form $n+1=ab$; 
with $2 < a,b$. 
Also assume that one of the following occures: 


*

*If $a= b$; then $5 \leq a=b$.

*If $a=2b$; then $5 \leq b  $.

*If $2a=b$; then $5 \leq a  $.

With the condition as in 
$\color{Blue}{\star}$ ; 
we claim that 
the assertion is $\color{Blue}{\text{true}}.$ 

Let $n, a, b$ to be fixed as above in $\color{Blue}{\star}$ . 
Notice that if we let $x:=a-1$ and $y:=b-1$ 
then one can easilly see that: 
$$n=xy+x+y.$$ 


$\color{Blue}{\text{Remark(I)}}$ : 
With the notation as in $\color{Blue}{\star}$; 


*

*If $a=b$ , with $5 \leq a$ 
then we have: 
$$b < 2b < 3b < 4b < ab=n+1 \Longrightarrow b < 2b < 3b <4b \leq n=ab-1 \ ; $$ 
so we can conclede that: 
$$(b)(2b)(3b)(4b)=24(ab)^2=24(n+1)^2 \mid n! ;$$ 
which implies that: $(n+1)^2 \mid n! \ \ $ . 





*

*If $a=2b$ ;  with $5 \leq b$ 
then we have: 
$$b < 2b < 3b < 4b < ab=n+1 \Longrightarrow b < 2b < 3b <4b \leq n=ab-1 \ ; $$ 
so we can conclede that: 
$$(b)(2b)(3b)(4b)=6(ab)^2=6(n+1)^2 \mid n! ;$$ 
which implies that: $(n+1)^2 \mid n! \ \ $ . 





*

*If $2a=b$ ;  with $5 \leq a$ 
then we have: 
$$a < 2a < 3a < 4a < ba=n+1 \Longrightarrow a < 2a < 3a <4a \leq n=ba-1 \ ; $$ 
so we can conclede that: 
$$(a)(2a)(3a)(4a)=6(ab)^2=6(n+1)^2 \mid n! ;$$ 
which implies that: $(n+1)^2 \mid n! \ \ $ .  





*

*If $a \neq b$ and $a \neq 2b$ and $2a \neq b$ ; 
then we have: 
$$b < 2b < ab=n+1 \Longrightarrow b < 2b \leq n=ab-1 \ ; $$ 
$$a < 2a < ab=n+1 \Longrightarrow a < 2a \leq n=ab-1 \ ; $$
so we can conclede that: 
$$(b)(2b)(a)(2a)=4(ab)^2=4(n+1)^2 \mid n! ;$$ 
which implies that: $(n+1)^2 \mid n! \ \ $ .  








Remark(II): Integer $N$ 
can not be written 
in the form $N=ab$; 
with $2 < a,b$; 
if and only if 
$N$ is prime or twice a prime. 

Remark(III): 
We can split the integres in three cases:


*

*$\color{Red}{   n+1 =   \text{prime numbers and twice of prime numbers; 
for which the assertion is wrong!}   }.$ 

*$\color{Purple}{ n+1= a.a 
 \ \ \ 
\text{for}    
\ \ \ 
a \in \{ 1, 2, 3, 4 \} }.$ 
This case has only four numbers; 
which can you check out that 
the assertion is $\color{Blue}{\text{true}}$ only for 
$\color{Blue}{n+1=16}$ and 
$\color{Blue}{n+1= 1}$ . 

*$\color{Purple}{ n+1= a.(2a) 
 \ \ \ 
\text{for}    
\ \ \ 
a \in \{ 1, 2, 3, 4 \} }.$ 
This case has only four numbers; 
which can you check out that 
the assertion is $\color{Blue}{\text{true}}$ only for 
$\color{Blue}{n+1=18}$ and 
$\color{Blue}{n+1=32}$ . 

$\color{Green}{\text{Remark(IV)}}$ : 
Except for the following cases; 
for which the assertion is 
$\color{Red}{\text{wrong}}$ 
1.$\color{Red}{   n+1 =   \text{prime numbers and twice of prime numbers 
}   }.$ 
2.$\color{Red}{ n+1= 4, 9, 2, 8 } .$ 
for all the other cases 
the assertion is 
$\color{Blue}{\text{true}}$ . 






$\color{Red}{\text{Remark(V)}}$ : 
Let $p$ to be an odd prime number; 
then we have: 
$p^2 \color{Red}{\nmid} (2p-1)! \ \ . $ 

For example let $p$ to be an odd prime number; 
and let's define: 
$$n:=2p-1 
\ \ \ 
\text{and} 
\ \ \ 
x:=1 
\ \ \ 
\text{and} 
\ \ \ 
y:=p-1.$$
Notice that we have: $n=xy+x+y$

We calaim that $(2p)^2 \color{Red}{\nmid} (2p-1)!$. 
[ 
Suppose on contrary 
that $(2p)^2 \mid (2p-1)!$ , 
on the otherhand notice that 
$p^2 \mid (2p)^2$. 
So we must have 
$p^2 \mid (2p-1)!$ 
; 
which has an obvious contradiction 
with the above $\color{Red}{\text{Remark(V)}}$. 
] 
</s>