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## Friday, December 31, 2010

### A conjecture about perfect numbers

#### Perfect number

In number theory the sum of the divisors is denoted as $\sigma$: $$\sigma(n) = \Sigma_{d/n} d$$ and $s(n)=\sigma(n) - n$ is the sum of the proper divisors. A perfect number is equal to the sum of its proper divisors. All known perfect numbers are even, it is unknown if odd perfect numbers exist. The number $2^{p-1}(2^p-1)$ is perfect if and only if $(2^p-1)$ is prime.

#### Conjecture

Show that: if $p$ is odd then
$$2^{p-1}(2^p-1) = \sum_{k=1}^{\frac{p+1}{2}-1} (2k-1)^3$$
( Notice that $2^{p-1}(2^p-1)$ yields a perfect number if $(2^p-1)$ is prime. )

#### Example

$6$ is perfect, since $6 = 1 + 2 + 3.$
$28$ is perfect, since $28= 1 + 2 + 4 + 7 + 14.$

Any perfect number ( except 6 ) can be represented as a sum of cubes.
$\begin{array}{ccc} \underline{p} & \underline{Pf} &\underline{s}\\ 3 & 28 & 1^3 + 3^3 \\ 5 & 496 & 1^3 + 3^3 + 5^3 + 7^3 \\ 7 & 8128 & 1^3 + 3^3 + ... + 15^3 \\ 13 & 33550336 & 1^3 + 3^3 + ... + 127^3 \end{array}$

#### Proof

My exercise for New Year's Day. ( You may have noticed that I like doing 'sums'. ) Later...

( Source:
- A primer of analytic number theory, From Pythagoras to Riemann by Jeffrey Stopple
)

## 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.)