Prove that perfect numbers are even

Definition of perfect number

Perfect number （Perfect number）, Also called perfect number or perfect number , It's some special natural number . All its real factors （ That is, a divisor other than itself ） And （ Factor function ）, Exactly equal to itself .
for example ：$1+2+3=6$

prove

If the complete numbers are even , That is to say, the perfect number is not odd
Suppose there are odd numbers that are perfect , Set it as $x$
set up $x$ The set of factors is $X$ ,$x$ The set of true factors of is $X^{\prime }$,$X^{\prime }=X-\{x\}$ ,$y=\sum X^{\prime }$ , Odd set is $U$ , Then there are $X\subset U$

take $U$ A subset of $U_n$ , And $n\in U,n\ge 3$ , among $U_n=\{1,3,5,...,n\},X\subset U_n$
because $x=1$ when $X^{\prime }$ For an empty set , So this time $x$ Not a perfect number , So from $n\ge 3$ To discuss
For a given set $X$ Yes $x=\prod X$
if $x>y$, hypothesis $x\cdot k\le y+k$ , be $k\le \frac{y}{x}+1<2$
here $\forall x_{sub}\in X,x_{sub}\ge 3,x_{sub}\cdot x>x_{sub}+y$, So for a given set $X$ Just need to discuss $x=\prod X$ The situation of

When $n=3$ when , Yes ：
$X=\{1\},x=1,X^{\prime }=\varnothing$, So this time $x$ Not a perfect number
$X=\{1,3\},x=3,X^{\prime }=\{1\},y=1,x>y$, So this time $x$ Not a perfect number
On the other hand $\forall X\subset U_3,x$ Not a perfect number
When $n=5$ when , Yes ：
$X=\{1,5\},x=5,X^{\prime }=\{1\},y=1,x>y$, So this time $x$ Not a perfect number
$X=\{1,3,5\},x=15,X^{\prime }=\{1,3,5\},y=9,x>y$, So this time $x$ Not a perfect number
On the other hand $\forall X\subset U_5,x$ Not a perfect number
Suppose that $n=k$ when , There is still a $\forall X\subset U_k,x_k>y_k$ , namely $x_k$ Not a perfect number
Then when $n=k+2$ when , Yes $U_{k+2}=U_k\cup \{k+2\}$
$\because x_k>y_k$ , And obviously $k+2>2$
$\therefore x_k\cdot (k+2)>y_k+(k+2)$
$\therefore$ Yes $\forall X\in U_{k+2}$ Yes $x_{k+2}>y_{k+2}$, namely $x_{k+2}$ Not a perfect number
$\therefore$ According to mathematical induction , Yes $\forall X\in U$ Yes $x>y$, namely $x$ Not a perfect number , That is, there is no odd number that is complete
$\therefore$ Perfect numbers are even numbers

explain

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