1.3 Fermat Primes, Mersenne Primes, and Perfect Numbers

Fermat claimed to have proven that the numbers
$F_k=2^{2^k}+1(k\ge 0)$
are all prime. The first 5 of these are prime, but it has since been shown that $F_5,...,F_{3}2$ are composite. The numbers $F_k$ are called Fermat Numbers and the prime ones are called Fermat Primes

For $p$ prime, denote $M_p=2^p-1$. Primes of this form are called Mersenne primes.

The sum-of-divisors function is given by
${\sigma }_1(n)={\sum }_{d\mid n}d$
where the sum is over positive divisors. A positive integer $n$ is perfect if ${\sigma }_1(n)=2n$

Example 1.3.1

We have $6=3+2+1$, so 6 is perfect.

An arithmetic function is a function $\mathbb{N}\to \mathbb{C}$. An arithmetic function $f$ is multiplicative if $f(mn)=f(m)f(n)$ holds for any coprime positive integers $m$ and $n$

Lemma 1.3.2

The sum-of-divisors function is multiplicative

Proof

Let $m,n\in \mathbb{N}$ with $(m,n)=1$. Then
${\sigma }_1(mn)={\sum }_{d\mid mn}d={\sum }_{d_1\mid m}{\sum }_{d_2\mid n}{d}_1{d}_2={\sigma }_1(m){\sigma }_1(n)$

Example 1.3.3

Let $M_p=2^p-1$ be a Mersenne prime, and put $n=2^{p-1}{M}_p$. Then
${\sigma }_1(n)={\sigma }_1(2^{p-1}){\sigma }_1(M_p)=(2^p-1)2^p=2n$
So $n$ is perfect.

Theorem 1.3.4 (Euclid-Euler Theorem)

If $n\in \mathbb{N}$ is even then $n$ is perfect if and only if
$n=2^{p-1}{M}_p$
where $M_p=2^p-1$ is a Mersenne Prime

Proof

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