Proposition: Even Perfect Numbers

The number $n:=\frac{p+1}2p=2^{k-1}(2^k-1)$ is an even perfect number, if and only if $p$ is a prime number having the form $p=2^k-1$ for an $k > 1, k\in\mathbb N.$

Examples

We calculate some different cases:

$k=2$:

$k=3$

$k=4$

$k > 1$, $k$ composite

$k=5$

Other cases and notes

Further perfect numbers can be found for the prime numbers $k=7\Rightarrow n=8128$, $k=13\Rightarrow n=33550336,$ etc.

Although perfect numbers and the above elementary result have been known for at least 2500 years (see Prop. 9.36 in Euclid's Elements), still unsolved mathematical problems are: * Are there infinitely many even perfect numbers. * Existence of odd perfect numbers.

Proofs: 1

  1. Proposition: Numbers Being the Product of Their Divisors

Propositions: 1


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927