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

I am a grad student studying mathematics at Emporia State University. I obtained a B.S. in Mathematics from the University of Memphis in 2010 where I served as President of the math club my senior year. My research interests include Graph Theory, Number Theory, Recreational Mathematics, Mathematical Physics, and Operator Theory.
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2 Responses to Odd Perfect Numbers: Do They Exist?

  1. Bob says:

    I’m leaving this post here for those who might stumble across you page in search of useful mathematical information regarding the Odd Per. # Conj.

    In your poor attempt to avoid plagiarism you completely misquoted correct mathematical information: In 1888, Eugène Charles Catalan proved that if an odd perfect number does exist it is not divisible by 3, 5, or 7 and has at least 26 prime factors (this result was later extended to 27 prime factors by K.K. Norton in 1960).

    The correct sentence is as follows: In 1888, Catalan proved that if an odd perfect number is not divisible by 3, 5, or 7, it has at least 26 distinct prime aliquot factors, and this was extended to 27 by Norton (1960).

    Catalan did not prove that Perf. Odd numbers cannot be divided by 3, 5, and 7.

    ~Bob

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