#### Question 19.5.1.

Does there exist an odd perfect number?

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

Does there exist an odd perfect number?

Yikes!

We do know some things about the question. Here are some fairly easy facts.

Here are simple forms of numbers that can’t be perfect.

- An odd perfect number cannot be a prime power.
- An odd perfect number cannot be a product of exactly
*two*prime powers. - An odd perfect number cannot be a product of exactly
*three*prime powers unless the first two are \(3^e\) and \(5^f\text{.}\)

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!

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 \(n\) and \(\sigma(n)\) are both odd, then \(n\) is a perfect square.

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.

If \(\frac{5}{3}\) is the abundancy index of \(N\text{,}\) then \(5N\) is an odd perfect number.

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.

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*}

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.

An odd perfect number must (as of 2021):

- Be greater than \(10^{1500}\text{.}\) (The most recent announcement
^{ 13 }says researchers have ‘pushed the computation to \(10^{2000}\)’, and you can help try to factor^{ 14 }some desired numbers to help compute up to \(10^{2100}\text{.}\)) - Have at least 101 prime factors (not necessarily distinct).
- Have at least 10
*distinct*prime factors. (This is new and relies on heavy computation by Pace Nielsen in Odd perfect numbers, Diophantine equations, and upper bounds in Mathematics of Computation^{ 15 }.) - Have a largest prime factor at least \(10^8\text{.}\)
- Have a second largest prime exceeding \(10000\text{.}\)
- Have the sum of the reciprocals of the
*prime divisors*of the number between*about*\(0.6\) and \(0.7\text{.}\) - Have the sum of the reciprocals of odd perfect numbers be finite (since the sum of the reciprocals of
*all*perfect numbers is finite!). In fact, the sum of the reciprocals of odd perfects must be less than \(2\times 10^{-150}\) (see [E.7.6]), and that of all perfects is less than about \(0.0205\text{.}\) - Obey the rule that if \(n\) is an odd perfect number, then \(n\equiv 1\text{ mod }12\) or \(n\equiv 9\text{ mod }36\text{.}\)

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.)

An odd perfect number must be of the form \(p^e m^2\text{,}\) where \(m\) is odd, \(p\) is prime, and \(p\) *and* \(e\) are both \(\equiv 1\text{ (mod }4)\text{.}\)

`www.lirmm.fr/~ochem/opn/`

There was another search at

`oddperfect.org`

but they seem to have let their domain lapse, so it is unclear whether it is still a going concern. (Search `web.archive.org`

for the status in 2019.)`www.lirmm.fr/~ochem/opn/`

`www.lirmm.fr/~ochem/opn/mwrb2100.txt`

`www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02941-X/`

`www.quantamagazine.org/mathematicians-open-a-new-front-on-an-ancient-number-problem-20200910/`

`eulerarchive.maa.org/hedi/HEDI-2006-11.pdf`