More about even perfect numbers

Knowing that every even perfect number is of the form

$n=2^ kp=2^ k(2^{k+1}-1)$

makes it easy to list all its factors, and this enables us to prove some interesting results. Since $p$ is a prime, the factors of $n=2^ kp$ must either be a power of $2$ (up to $2^ k$ ) or a product of a power of $2$ and $p$ . Listing these factors in a table we get:

Factors of $2^ k$	Other factors
$1$	$p$
$2$	$2p$
$2^2$	$2^2p$
$2^3$	$2^3p$
...	...
$2^{k-1}$	$2^{k-1}p$
$2^ k$	$2^ kp$

For example, the factors of $n=28=2^4 \times 7$ are:

Factors of $2^4$	Other factors
$1$	$7$
$2$	$2\times 7=14$
$2^2$	$2^2\times 7=28$

We claimed that the product of the factors of an even perfect number $n=2^ kp$ is always equal to $n^{k+1}$ . For $n=24$ this is clearly true, since

$1\times 2 \times 4 \times 7 \times 14 \times 28=21952=28^3,$

as required.

For a general proof, note that the product of the factors in the first column of the table above is

$\begin{equation} 1 \times 2 \times 2^2 ...\times 2^ k = 2^{1+2+...+k}.\end{equation}$

(1)

As we already noted in the main article, the sum of the first $k$ integers is

$1+2+..+k = k(k+1)/2,$

so (1) is equal to

$2^{1+2+...+k}=2^{k(k+1)/2}.$

Note that every factor in the second column of the table above is $p$ times the corresponding factor in the first table. Since there are $k+1$ entries in each column, the product of all the factors in the second column is therefore

$p^{k+1} 2^{k(k+1)/2}.$

The product of all the factors is

$2^{k(k+1)/2}p^ k 2^{k(k+1)/2}=p^{k+1}2 ^{k(k+1)}=(2^ k p)^{k+1}=n^{k+1}.$

This proves our result.

A similar result involves adding up the reciprocals of all the factors of an even perfect number. Taking our example of $n=28$ again we see that

$\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{7}+\frac{1}{14}+\frac{1}{28}=2.$

In fact the sum is equal to $2$ for every perfect number! As one might expect, the proof proceeds in the same way as above, replacing addition with multiplication. We leave it to you as an exercise.

Back to the main article

Add new comment