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## Saturday, January 1, 2011

### A conjecture about perfect numbers ( - continued )

#### 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. )

#### Plan

The plan of the proof is as follows.
- Find expressions for $1^3 + 2^3 + 3^3 + ... + k^3$
- and $2^3 + 4^3 + 6^3 + ... + (2k)^3$
- Subtract both expressions
- Create $f(p)$ by injecting $2^p-1$ into the upper-index

#### Proof

We use the Pascal Triangle to determine $\sum_{k=1}^{n} k^3$.

$\underline{n}$
0: 1
1: 1 - 1
2: 1 - 2 - 1
3: 1 - 3 - 3 - 1
4: 1 - 4 - 6 - 4 - 1
5: 1 - 5 - 10-10 - 5 - 1

Since $n^3={n \choose 1} + 6{n \choose 2} + 6{n \choose 3}$ we seek the second column, one row down or $\sum_{k=1}^{n} k^3 = {n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 3}$.

Clearly, $\sum_{k=1}^{n} 2k^3 = 8({n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4})$.

If
$$f(n)=\sum_{k=1}^{n} k^3 = {n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4}$$
and
$$g(n)=8 \sum_{k=1}^{n} k^3 =8({n+1 \choose 2} + 6{n+1 \choose 3} + 6{n+1 \choose 4})$$
then the function we require is $$s(n) = f(n) - g( \lfloor \frac{n}{2} \rfloor ) \text{ for } n=1,3, \cdots$$.
$$\begin{array}{ll} \underline{n} & \underline{s(n)}\\ 1 & 1 \\ 3 & 28 \\ 5 & 153 \\ 7 & 496 \\ 9 & 1225 \\ 11 & 2556 \\ 13 & 4753 \\ 15 & 8128 \end{array}$$
Since $n=1,3, \cdots$ anyway, we can remove the difficult to handle floor function by $\frac{n-1}{2}$ giving $$s(n) = f(n) - g(\frac{n-1}{2}) \text{ for } n=1,3, \cdots .$$
Finally we rework
$$s(p)=f(2^{\frac{p+1}{2}}-1) - g(\frac{(2^{\frac{p+1}{2}}-1)-1}{2})$$
to $$s(p)=2^{p-1} \left(2^p-1\right)$$
yielding for $p=1 \cdots 7$
$$\begin{array}{ll} \underline{p} & \underline{s(p)}\\ 1. & 1. \\ 2. & 6. \\ 3. & 28. \\ 4. & 120. \\ 5. & 496. \\ 6. & 2016. \\ 7. & 8128. \end{array}$$
( Since $2^3-1$, $2^5-1$ and $2^7-1$ are prime $28$ and $496$, $8128$ are perfect. )

## 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.)