# Theorem of Even Perfect Numbers/Necessary Condition

## 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.

## Proof

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 | |||||||||||

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

$\blacksquare$

## Historical Note

René Descartes stated in $1638$ that he had a proof that every even perfect number is of the form $2^{p - 1} \paren {2^p - 1}$, but failed to actually produce it.

The first actual published proof was made by Leonhard Paul Euler.

## Sources

- 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Conjecture $2$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $28$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.2$: More about Numbers: Irrationals, Perfect Numbers and Mersenne Primes - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $28$