What is particularly interesting about this is the connection to something we silently omitted until now. This is the question whether there are odd perfect numbers! The connection below is due to P. Weiner. We begin with a useful lemma.

However, we don't know whether the hypothesis is ever true. In fact, we don't even know the answer to the following:

Open Question: Does there exist an odd perfect number?

Yikes! This question is still open after two and a half millennia. We do know some things about the question, though. First, recall that \[\frac{\sigma(n)}{n}=\prod_{i=1}^k\frac{p_i-1/p_i^{e_i}}{p_i-1}< \prod_{i=1}^k\frac{p_i}{p_i-1}\] when \(n\) is a product of the prime powers \(p_i^{e_i}\).

Of course, this is pretty 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 two big searches going on right now.

• Must be greater than \(10^{1500}\). (The proof of this is now nearly complete, at any rate; slightly smaller powers like 1250 are confirmed.)
• Has at least 101 prime factors (not necessarily distinct).
• At least 9 distinct prime factors.
• Largest prime factor must be at least \(10^8\).
• Second largest prime exceeds \(10000\).
• The sum of the reciprocals of the prime divisors of the number is between about \(0.6\) and \(0.7\).
• The sum of the reciprocals of all perfect numbers is finite, so the sum of the reciprocals of odd perfect numbers definitely is!
• If \(n\) is an odd perfect number, then \(n\equiv 1\text{ mod }12\) or \(n\equiv 9\text{ mod }36\).

Finally, here is the link to an article about Euler and his own criterion.

An appropriate way to finish up this at times overwhelming overview, since he finished the characterization of even perfect numbers.