# Prove that an odd perfect number has 3 distinct primes

So far i've tried looking at an equation which is true for perfect numbers:

$\sigma(n) = 2n$. where $\sigma(n) = \sum_{d|n} d$

i started looking at a odd perfect number which has 1 prime divisor say $p$, then:

$\sigma(n) = \sum_{i \in \mathbb{N}} p_i$, where all $p_i$'s are the same except for $p_1 = 1$. But then $\frac{\sigma(n)}{n} = 2$ right...?

Any helpfull tips here?

Kees

## 1 Answer

Hints:

(1) $\sigma(p^a)=1+p+p^2+...+p^a=\frac{p^{a+1}-1}{p-1}$. And so $\frac{\sigma(p^a)}{p^a}<\frac{p}{p-1}$.

(2) So $\frac{\sigma(p^a\cdot q^b)}{p^a\cdot q^b}<\frac{p}{p-1}\cdot\frac{q}{q-1}$

(3) $\frac{p}{p-1}$ is decreasing (i.e., it is larger for small primes).

• nice, i think i get it but try to understand why the last inequality of (1) is true if $p^\alpha$ is odd... – Kees Til Sep 8 '15 at 16:10
• @KeesTil The inequality in (2)? Or... – paw88789 Sep 8 '15 at 16:12
• i especially mean is this true for even perfect numbers as well? – Kees Til Sep 8 '15 at 16:12
• The inequalities hold also for $p=2$. – paw88789 Sep 8 '15 at 16:13
• $\frac{p^{a+1}-1}{p^a (p-1)}=\frac{p}{p-1}-\frac{1}{p^a (p-1)}<\frac{p}{p-1}$ – paw88789 Sep 8 '15 at 16:27