Roots of Unity

Roots of Unity

Mathematics: learning it, doing it, celebrating it.

Where in the World Are the Odd Perfect Numbers?


A ladybug with a perfect number of spots. Image: Lauren Tucker Photography, via flickr.

The launch of Roots of Unity was just in time for this year's Joint Mathematics Meetings, a mathapalooza put on by the two largest professional mathematical societies (the American Mathematical Society and the Mathematical Association of America). Nearly 6,000 of my closest mathematical friends are here with me in sunny San Diego taking in the lectures, panel discussions, poster sessions, social events, and art gallery. On Wednesday, the first day of the meetings, I attended talks about using art in the mathematics classroom, modeling atmospheric and ocean currents, celestial mechanics, and WWII codebreakers, but this post is about a talk I did not attend. I was in the assigned room at the time the talk should have happened, but the speaker was not.

The talk was titled "Some Recent Results on Odd Perfect Numbers." A number is called perfect if it is the sum of its positive factors other than itself. For example, 6=3+2+1, and 3, 2, and 1 are the factors of 6. The next two perfect numbers are 28 and 496, and so far only 47* perfect numbers have been discovered. (Numbers that aren't perfect are called either deficient or abundant, depending on whether the sum of factors is smaller or larger than the number.) I am not a number theorist, but I have been fascinated by perfect numbers since I was a little kid and my dad told me that he and my mom got married on the 28th of the month because 28 is a perfect number.

One interesting aspect of perfect numbers is their connection to a certain type of prime number called a Mersenne prime. Mersenne primes are prime numbers that are one less than a power of two, so they can be written 2n-1 for some number n. The number 3, for instance, is a Mersenne prime because it can be written 22-1. Euclid, sometimes called the father of geometry, proved that when 2n-1 is prime, the number (2n-1)(2n-1) is a perfect number, and over 2000 years later, Swiss mathematician Leonhard Euler proved that all even perfect numbers have the form (2n-1)(2n-1). The perfect number 6=2×3 has this property for n=2, and you can check 28 and 496 for yourself.

So the even perfect numbers are more or less sorted out, but what about the odd ones? Well, no one knows a single one, or whether any can exist. Which is why it was quite amusing to me (and several other would-be audience members) that the speaker did not show up to the session on odd perfect numbers. I checked the abstract booklet just in case we were all part of a prank or piece of abstract performance art, and that doesn't appear to be the case. In part, the abstract said, "In 1991, Brent, Cohen, and te Riele proved that odd perfect numbers are greater than 10300. In 2012, Ochem and Rao modified their method to show that odd perfect numbers are greater than 101500. Some recent results on odd perfect numbers will be discussed in this presentation."

I am not familiar with the techniques number theorists use to study perfect numbers, so I can't speculate on what new results the speaker would have presented, but I found it poetic that of all talks I might attend for which the speaker would fail to materialize, it would be the one about a possibly empty set of numbers.

I shouldn't speculate about why the speaker didn't show up to the talk either, but I will anyway. The meetings are in San Diego, so Carmen Sandiego kidnapped the speaker and all the odd perfect numbers. Gumshoes, it's up to you to figure out where she's hiding them! V.I.L.E. henchman Ruth Less has a clue for you...

*Addendum February 5, 2013: this sentence was correct for 15 days. On January 25, 2013, a new Mersenne prime, and hence a new perfect number, was discovered by the GIMPS project, bringing us to a grand total of 48 of each.

The views expressed are those of the author and are not necessarily those of Scientific American.

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