Odd Perfect Numbers: Do They Exist?

images-2Mathematical 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.


About Avery Carr

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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35 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.

    • A H Welch says:

      I agree with the second statement. And if the second statement is true, why would anyone conclude that the set of odd perfect numbers would be likely to have at least one element up to 10^300 (or even 10^1500)?

  3. Avery Carr says:

    Late I know, but thank you for your comments.

  4. Şaban YENER says:

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

  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?

    • Jenaro Tomaszewski says:

      198585576189. It is the only known odd number that would be perfect if one of its composite divisors was counted as a prime

  10. Matthias says:

    I found the proof. Just double checking my calculations. But the MAX character settings of comments section of this blog is too small to contain my proof.

  11. Mohammed Ijas says:

    Odd perfect number does not exist.
    According to Euclid’s formulae a perfect number is = 2^P(2^P-1) where (2^P-1) should be prime. Only 2 is the even prime number and rest are odd. So (2^P-1) is odd. (2^P-1) +1= 2^P
    That is (2^P-1) is odd , an odd number + 1 is even . Therefore 2^P is even.
    Product of odd and even is always even.
    Since (2^P-1) is odd and 2^P is even 2^P(2^P-1) is even.
    That is odd perfect number does not exist.

    • Matthew Sun says:

      Well, of course. We know that already. What we are trying to find out here is if there exist any perfect numbers that fall outside of Euclid’s form of a perfect number, and therefore odd. I know I’m not good enough with number theory to figure that out, but perhaps someone can.

    • A H Welch says:

      The comment starts out with an early incorrect statement and does not improve. Is this an intentional attempt at humor?

    • Ankur Tagra says:

      If some one can prove or disprove that odd prime numbers does not exist, how can they publish their result ?

    • Pedro Fernandez says:

      Your proof is FALSE since all the perfect numbers of the form 2^(p-1)(2^p-1) , where ‘p’ is prime and (2^p-1) are Mersenne prime numbers, are EVEN perfect numbers. So the condition of Mersenne prime for a perfect number be odd it is NOT necessary.

    • Logan says:

      his proof states that those numbers are indeed perfect numbers. His proof DOES NOT state that all perfect numbers fits that criteria

  12. John Cena says:

    No odd numbers have 2 in their prime factorization. That means their entire prime factorization is odd. There is no combination of of prime numbers (without 2) that add up to an even or prime number. If The factorization adds up to an even number, it can’t be an odd perfect number. If they add up to a prime, they for sure aren’t factors of that prime.

  13. KirK says:

    “one may recognize that if $p$ and $2^p-1$ are prime, then $2^p-1$ is a Mersenne Prime” — this statement is true, but stated a bit confusingly. By usual definition, a prime of the form $2^p-1$ is called a Mersenne Prime, and naturally, from $2^p-1$ being prime it follows that $p$ is prime (since if, conversely, $p=m n$ then $2^{m n}-1$ is divisible by $2^{n}-1$ and $2^{m}-1$ and thus not prime). So $p$ being prime is a consequence of the number being a Mersenne prime, not a precondition for it, as seems to be stated.

  14. Harshad says:

    Any perfect number cannot be a perfect square. It’s an obvious proof by the definition. Also sum of reciprocals of divisors of a perfect number is always 2. This is another obvious results from the definition of perfect numbers.

  15. Killian Heanue says:

    I have a proof of this, it is currently being analysed and published.

  16. Mirwais Fawad Lodin says:

    I would describe this as overexaggerating my solution. Anyone should be able to solve this.

  17. Steven ELmore says:

    All these comments have gone through my brain since being introduced to Odd Perfect Numbers in 1975. I helped Dr. Beauregard Stubblefield with software in his proof that there is no odd perfect number less than 10^100 (www.glerl.noaa.gov/ftp/publications/tech_reports/Stubblefield_NOAA_ERL_1977.pdf). There are two main points that I would like good folks to consider.
    First, even perfect numbers are explained simply: If p and (2^p – 1) are prime, then 2^{p-1} * (2^p – 1) is perfect.
    There are simple rules for Odd Perfect Numbers also, although this has not been published quite so plainly. Suppose N is an Odd perfect Number. Then (1) N is odd, (2) N is perfect, and (3) N is less than a bound, call it M. Ochem and Rao set M = 10^{1500} and could not find an Odd Perfect Number. But that is the point, and it made them famous.
    The second item is hard to swallow. All mathematicians know that an Odd Perfect Number does not exist. So why the hullabaloo? The point is not about Odd Perfect Numbers themselves but about the mathematics that has been discovered in the process. It is about the interest and excitement generated by one of the oldest unproven propositions in mathematics. It is about drawing people to math.

  18. Chris Ferrante says:

    I was able to prove that if Q is an odd perfect number, then Q must be of the form:

    Q = p^n * m^2, where p, n, and m are positive integers such that:

    (a) p = 1 mod 4 and prime
    (b) n = 1 mod 4
    (c) m is not divisible by p or 2

    However, after more research I found out Euler proved it first. Every time I come up with something interesting, I find out Euler proved it first.

  19. Alan Caldwell says:

    There are no perfect odd numbers because there is no reason for them

    • Steven Elmore says:

      There is no reason for factoring large numbers, so why find ways of factoring them? Oh, wait. Our entire internet security is based on large primes and our inability to factor their products.

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