# Theorem of Even Perfect Numbers

## Theorem

Let $a \in \N$ be an even perfect number.

Then $a$ is in the form:

- $2^{n - 1} \paren {2^n - 1}$

where $2^n - 1$ is prime.

Similarly, if $2^n - 1$ is prime, then $2^{n - 1} \paren {2^n - 1}$ is perfect.

## Proof

### Sufficient Condition

Suppose $2^n - 1$ is prime.

Let $a = 2^{n - 1} \paren {2^n - 1}$.

Then $n \ge 2$ which means $2^{n - 1}$ is even and hence so is $a = 2^{n - 1} \paren {2^n - 1}$.

Note that $2^n - 1$ is odd.

Since all divisors (except $1$) of $2^{n - 1}$ are even it follows that $2^{n - 1}$ and $2^n - 1$ are coprime.

Let $\map {\sigma_1} n$ be the divisor sum of $n$, that is, the sum of all divisors of $n$ (including $n$).

From Divisor Sum Function is Multiplicative, it follows that $\map {\sigma_1} a = \map {\sigma_1} {2^{n - 1} } \map {\sigma_1} {2^n - 1}$.

But as $2^n - 1$ is prime, $\map {\sigma_1} {2^n - 1} = 2^n$ from Divisor Sum of Prime Number.

Then we have that $\map {\sigma_1} {2^{n - 1} } = 2^n - 1$ from Divisor Sum of Power of Prime.

Hence it follows that $\map {\sigma_1} a = \paren {2^n - 1} 2^n = 2 a$.

Hence from the definition of perfect number it follows that $2^{n - 1} \paren {2^n - 1}$ is perfect.

$\Box$

### Necessary Condition

Let $a \in \N$ be an even perfect number.

We can extract the highest power of $2$ out of $a$ that we can, and write $a$ in the form:

- $a = m 2^{n - 1}$

where $n \ge 2$ and $m$ is odd.

Since $a$ is perfect and therefore $\map {\sigma_1} a = 2 a$:

\(\ds m 2^n\) | \(=\) | \(\ds 2 a\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {\sigma_1} a\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {\sigma_1} {m 2^{n - 1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \map {\sigma_1} m \map {\sigma_1} {2^{n - 1} }\) | Divisor Sum Function is Multiplicative | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {\sigma_1} m {2^n - 1}\) | Divisor Sum of Power of Prime |

So:

- $\map {\sigma_1} m = \dfrac {m 2^n} {2^n - 1}$

But $\map {\sigma_1} m$ is an integer and so $2^n - 1$ divides $m 2^n$.

From Consecutive Integers are Coprime, $2^n$ and $2^n - 1$ are coprime.

So from Euclid's Lemma $2^n - 1$ divides $m$.

Thus $\dfrac m {2^n - 1}$ divides $m$.

Since $2^n - 1 \ge 3$ it follows that:

- $\dfrac m {2^n - 1} < m$

Now we can express $\map {\sigma_1} m$ as:

- $\map {\sigma_1} m = \dfrac {m 2^n} {2^n - 1} = m + \dfrac m {2^n - 1}$

This means that the sum of all the divisors of $m$ is equal to $m$ itself plus one other divisor of $m$.

Hence $m$ must have exactly two divisors, so it must be prime by definition.

This means that the other divisor of $m$, apart from $m$ itself, must be $1$.

That is:

- $\dfrac m {2^n - 1} = 1$

Hence the result.

$\blacksquare$

## Comment

Hence, the hunt for even perfect numbers reduces to the hunt for prime numbers of the form $2^n - 1$.

From Primes of form Power Less One, we see that for $2^n - 1$ to be prime, $n$ itself must be prime.

See Mersenne prime.

## Also see

## Historical Note

The first part of this proof of the **Theorem of Even Perfect Numbers** was documented by Euclid in *The Elements*: Proposition $36$ of Book $\text{IX} $: Theorem of Even Perfect Numbers/Sufficient Condition.

The second part was achieved by Euler.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Mersenne numbers** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Mersenne numbers** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**Mersenne prime** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**perfect number** - 2021: Richard Earl and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(6th ed.) ... (previous) ... (next):**perfect number**