Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers with a bizarre property that they can be formed by adding up all the smaller numbers that make up their divisors.

The number $6$ is the first perfect number, because it may be divided by $1$, $2$ and $3$, and these numbers add up to $6$. Not many numbers are perfect.

The number $8$ is not, for $1$, $2$ and $4$ only add up to $7$.
that is, $1+2+4 \ne 8$

Nor is the number $9$, for $1$ and $3$ only add to $4$.
That is, $1+3 \ne 9$

Those four are:

\( \begin{align} \displaystyle
6 &= 1 + 2 + 3 \\
28 &= 1 + 2 + 4 + 7 + 14 \\
&= 1 + 2 + 3 + 4 + 5 + 6 + 7 \\
&= 1^3 + 3^3 \\
496 &= 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 \\
&= 1 + 2 + 3 + \cdots + 29 + 30 + 31 \\
&= 1^3 + 3^3 + 5^3 + 7^3 \\
8128 &= 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 \\
&= 1 + 2 + 3 + \cdots + 125 + 126 + 127 \\
&= 1^3 + 3^3 + 5^3 + 7^3 + 9^3 + 11^3 + 13^3 + 15^3 \\
\end{align} \)

Euclid proved that $2^{p−1}(2^p − 1)$ is an even perfect number whenever $2^p − 1$ is prime.

For example, the first four perfect numbers are generated by the formula $2^{p−1}(2^p − 1)$, with $p$ a prime number, as follows:

for $p = 2$: $2^1(2^2 − 1) = 6$
for $p = 3$: $2^2(2^3 − 1) = 28$
for $p = 5$: $2^4(2^5 − 1) = 496$
for $p = 7$: $2^6(2^7 − 1) = 8128$

By the time we reach the twentieth perfect number its size has become unthinkably huge. The twentieth perfect number has an astonishing $5834$ digits. The first four perfect numbers have been known for over two thousand years.

Their significance has been debated since their discovery. When mathematics, philosophy and religion were all aspects of the same thing, it was perhaps natural for people to believe that perfect numbers must be favoured by God.

For example, Saint Augustine (354-430) wrote in his famous text $\textit{The City of God}$.

‘Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect…’

Likewise, $28$ was thought to be chosen by God as the perfect number of days to take the Moon to orbit the Earth, although today we know that the number is closer to $27.322\cdots$ days.

But perhaps Rene Descartes summarised perfect numbers best when he said:
‘$\textit{Perfect numbers like perfect men are very rare}$’.

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