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## Friday, February 25, 2011

### Perfect numbers

An example of a note in my NT Wiki. In M381 only the first part of the proof is given which was known as early as Euclid. Euler was the first who gave, a not so very clear proof of the reverse. Many followed Euler with subsequent improvements of the proof. I have used a proof of Dickson published in 1911. - Perfect numbers are still actively researched. It is for example still unknown if odd perfect numbers exist. If they exist however they are surely very large.

An even integer is perfect if and only if it can be written as $2^{p-1}(2^p-1)$, where both $p$ and $2^p-1$ are prime.

We show that:
If $n = 2^{p-1}(2^p-1)$ and $p$ and $2^p-1$ are prime then $n$ is perfect.
\begin{align}
\sigma(n) &= \sigma(2^{p-1}(2^p-1)) \\
&= \sigma(2^{p-1}) \sigma(2^p-1) \ \\
& = ( 2^p-1 ) 2^p \\
& = 2^p ( 2^p-1 ) \\
& = 2 (2^{p-1}(2^p-1)) \\
& = 2n
\end{align}
Since $\sigma(n)=2n$ we conclude that $n$ is perfect.

We show that:
If $n$ is even and perfect then it can be represented as $n=2^{p-1}(2^p-1)$ where $p$ and $2^p-1$ are prime.

Since $n$ is even we assume $n=2^{k-1}m$ where $(2^{k-1}, m) = 1$.
(1) We calculate the divisor-sum of $n$ as follows:
\begin{align}
\sigma(n) &= \sigma(2^{k-1}m)\\
& = \sigma(2^{k-1})\sigma(m)\\
& = \frac{2^k-1}{2-1}\sigma(n).
\end{align}

(2) We assume $n$ is perfect thus:
\begin{align}
\sigma(n) &= 2n\\
& = 2 (2^{k-1} m)\\
& = 2^k m
\end{align}

Now (1) and (2) gives:
\begin{align}
\frac{2^k-1}{2-1}\sigma(m) & = 2^k m \Leftrightarrow \\
\sigma(m) & = \frac{2^k m}{2^k-1} \\
& = \frac{((2^k-1)+1)m}{2^k-1} \\
& = m + \frac{m}{2^k-1}
\end{align}

Since $\sigma(m)$ is the sum of all divisors $\frac{m}{2^k-1}$ must be $1$. So $m=2^k-1$ with divisors $1$ and $m$ and is thus prime.

We conclude that $n=2^{k-1}(2^k-1)$ with $2^k-1$ and $k$ prime.
QED

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)