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Section 19.5 Odd Perfect Numbers

Subsection 19.5.1 Are there odd perfect numbers?

Let’s return to a question alluded to earlier -- one whose answer is still unknown after two and a half millennia:

Question 19.5.1.

Does there exist an odd perfect number?
We do know some things about the question. Here are some fairly easy facts.
We leave many details to Exercise 19.6.24. The easiest way to approach this is by cases and subcases, using the computation from Section 19.3 that
\begin{equation*} \frac{\sigma(n)}{n}=\prod_{i=1}^k\frac{p_i-1/p_i^{e_i}}{p_i-1}< \prod_{i=1}^k\frac{p_i}{p_i-1} \end{equation*}
when \(n\) is a product of the prime powers \(p_i^{e_i}\text{.}\)
  • An odd perfect number cannot be a prime power. This is easy; using the computation for \(k=1\) would require \(2=\frac{\sigma(n)}{n}<\frac{p}{p-1}\text{.}\) Even for \(p=2\text{,}\) \(2< p/(p-1)\) isn’t possible; since we are looking for an odd perfect number, it definitely won’t be possible!
  • An odd perfect number cannot be a product of exactly two prime powers. Use the same idea, but now with the biggest possible values for odd primes.
  • An odd perfect number cannot be a product of exactly three prime powers unless the first two are \(3^e\) and \(5^f\text{.}\) This proof is slightly longer.
    • Suppose that \(3\) is not the smallest prime involved. Then the biggest that
      \begin{equation*} \frac{p_1}{p_1-1}\cdot \frac{p_2}{p_2-1}\cdot \frac{p_3}{p_3-1} \end{equation*}
      can be is
      \begin{equation*} \frac{5}{4}\cdot \frac{7}{6}\cdot \frac{11}{10}=\frac{77}{48} \end{equation*}
      and this fraction is still less than \(2\text{.}\)
    • Suppose that \(5\) is not the second-smallest prime involved (assuming \(3\) is the smallest). We again get a contradiction.
This proof is from [E.2.8, Section 3.3A], which has even more details – including a full elementary proof that an odd perfect number must have four different prime factors!

Subsection 19.5.2 The abundancy index and odd perfect numbers

What is particularly interesting about this is that we can connect odd perfect numbers to a non-integer abundancy index in a surprising way! The connection below is due to P. Weiner in [E.7.14].
We begin with a useful lemma, which answers questions very closely related to Exercises 19.6.11 and 19.6.12.
If \(n\) is odd, it is a product of odd prime powers. Let’s look at \(\sigma\) as applied to each piece, thanks to multiplicativity.
If \(\sigma(n)\) is odd, then each factor \(1+p+p^2+\cdots +p^e\) is odd. Such a factor of \(\sigma(n)\) is a sum of odd numbers, which is only odd if there is an odd number of them.
Since there are \(e+1\) summands, \(e\) must be even for every primes \(p\) dividing \(n\text{.,}\) which finishes proving the lemma.
Assume this works for some \(N\text{.}\) Then \(3\sigma(N)=5N\text{.}\)
Let’s look at divisors. First, \(3\mid N\text{.}\) So if \(N\) is even, then \(6\mid N\text{,}\) so by Fact 19.4.10,
\begin{equation*} \sigma_{-1}(N)\geq \sigma_{-1}(6)=2>\frac{5}{3}\text{,} \end{equation*}
which is impossible. If \(N\) is not even, then \(N\) is odd, so \(3\sigma(N)=5N\) is odd, which implies \(\sigma(N)\) itself is odd.
Since \(3\mid N\) and using Lemma 19.5.3, we see that we must have that \(3^2\mid N\text{.}\)
Let’s return to the divisors. We know that \(5\nmid N\text{,}\) because otherwise
\begin{equation*} \sigma_{-1}(N)\geq \sigma_{-1}\left(3^2\cdot 5\right)=\frac{26}{15}>\frac{5}{3} \end{equation*}
which is again impossible.
Now we can compute directly that
\begin{equation*} \sigma_{-1}(5N)=\sigma_{-1}(5)\sigma_{-1}(N)=\frac{6}{5}\frac{5}{3}=2\; ! \end{equation*}

Subsection 19.5.3 Even more about odd perfect numbers, if they exist

Naturally, all of this is somewhat elementary; there are many more criteria. They keep on getting more complicated, so I can’t list them all, but here is a selection, including information from a big computer-assisted search 11  12  going on right now.
For another introduction to the problem focusing on ‘near-misses’/‘spoofs’, see this article in Quanta magazine 16 .
As an appropriate way to finish up this at times overwhelming overview, since Euler finished the characterization of even perfect numbers, let us present his own criterion for odd perfects! (See also the linked article 17  [E.7.19] by Euler expert Ed Sandifer.)
There was another search at but they seem to have let their domain lapse, so it is unclear whether it is still a going concern. (Search for the status in 2019.)