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 is prime, when $p$ is prime, then is a perfect number. From my last post on “The Infinitude of Mersenne Primes”, one may recognize that if $p$ and are prime, then 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 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 or .
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.