Odd Perfect Numbers: Do They Exist?

Mathematical inquiry can often lead to a jungle of unique questions and problems.  In the field of Number Theory, there are a wide assortment of such mathematical creatures.  Although these problems are easy to state, they can remain dormant for years with little sign of progress.   In fact, the Odd Perfect Number Conjecture is one such problem that has escaped proof for centuries.

Perfect numbers are positive integers that are the sum of their proper divisors.  For instance, 6 is a perfect number, because the sum of its proper divisors, 1, 2, and 3 equals 6 (1 + 2 + 3 = 6).  Euclid first devised a way to construct a set of even perfect numbers in Book IX of The Elements.  In his book, Euclid showed that if $2^p- 1$ is prime, when $p$ is prime, then $2^ {p-1} (2^p-1)$ is a perfect number.  From my last post on “The Infinitude of Mersenne Primes”, one may recognize that if $p$ and  $2^p-1$ are prime,  then $2^p-1$ is a Mersenne Prime.

In 1638, René Descartes sent a letter to Marin Mersenne stating that he believed every even perfect number is of Euclid’s form.  Furthermore, in the letter, Descartes was the first to reason that an odd perfect number may or may not exist.   Many mathematicians since have failed to produce a proof.  So, does there exist an odd perfect number?

Computationally the conjecture has been checked for odd numbers up to $10^{300}$  with no success.    Over time, mathematicians have produced several remarkable results.  In 1888, Eugène Charles Catalan proved that if an odd perfect number does exist and it is not divisible by 3, 5, or 7, then it has at least 26 prime factors (this result was later extended to 27 prime factors by K.K. Norton in 1960).  Another remarkable result came from the mathematician J. Touchard.    In 1953, Touchard showed that if an odd perfect number exists it must be of the form $12k+1$ or $36k + 9$.

Resources and more examples can be found easily on the internet.  The Norwegian mathematician Øystein Ore had the following to say about the conjecture and Euclid’s form in his book Invitation to Number Theory:

“This result shows that each Mersenne prime gives rise to a perfect number…. Are there any other types of perfect numbers?… This leaves us with the question: ARE THERE ANY ODD PERFECT NUMBERS?  Presently we know of none and it is one of the outstanding puzzles of number theory to determine  whether an odd perfect number can exist….”

From Ore’s words, the conjecture is definitely an outstanding puzzle.  Elegance is a word that mathematicians use when describing a result that is parsimonious and rigorous.  It would be nice to see an elegant solution to this old conundrum.  One that exhibits robustness and breeds more questions of like interest and  uniqueness.

Avery Carr is a senior analyst and past senior editor for the American Mathematical Society Grad Blog. He and his wife, Alison, live in Olive Branch, MS.
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13 Responses to Odd Perfect Numbers: Do They Exist?

1. Richard Hanlon says:

what a wonderful article! keep on rockin’, so to speak.

2. Jenaro Tomaszewski says:

As all perfect numbers up to 10^300 are even, it is unlikely that there are any odd perfect numbers. If there are any odd perfect numbers, then they are very rare.

• Jenaro Tomaszewski says:

I learned something else recently. An odd perfect number must not be divisible by 105.

• Jenaro Tomaszewski says:

I’m interested in large numbers. I know mathematicians like Jonathan Bowers & Sbiis Saibian. I heard that Jonathan Bowers coined 2 number names this year: oblivion & utter oblivion. These numbers are both larger than his famous meameamealokkapoowa oompa.

3. Avery Carr says:

4. Şaban YENER says:

Sayın authorized,
I think I found the perfect odd number.
What should I do?
Best regards,

• Hiro says:

Write it down.

5. Ct says:

Why isn’t the idea of an odd perfect number absurd on its face?

• Andreas Eisele says:

Why should it be absurd? If clearly impossible, one would have a proof. Obviously that has not been achieved so far, so it remains possible, but no such number has been found, so it remains an open problem. Do you perhaps mean it would be irrelevant? Maybe so, but it is also possible (actually likely) that a proof (in either direction) would reveal some other useful by-products, so the attempt might be worthwhile.

6. Michelle says:

Hello Dr. Math,
Thank you for your note about the prime numbers and perfact numbers. Follwing your explanations and questions I could step into a number theory and realized it is an interesting subject.

7. James N. Bridgeman says:

Proving that no odd perfect number exists seems very easy. Has anyone published a proof yet?
Also, has anyone published the proof of Fermat’s last theorem – the one that he would have written over 350 years ago, not the complex one published by Andrew Wilde?

8. Biranchi narayan dhal says:

I can’t get the proper answer….

9. David says:

Which odd number has come the closest to being a perfect number?