Multiperfect numbers (MPFNs)


13 May 1997: Ron Sorli (University of Technology, Sydney), using a program written by Jason Moxham (University of Southampton), has just discovered the first multiperfect number with index 10!

These pages aren't ready for prime time yet, although the glossary is in pretty good shape.

Just the basics:

As you may know, a perfect number is a positive integer which is equal to the sum of its proper divisors. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14. Number theorists have found that it is more convenient to consider the sum of all of the divisors of a number (because this function is multiplicative), and they call the sum of the divisors of N "sigma(N)". Thus N is perfect if and only if sigma(N) = 2N.

A multiperfect number (or "MPFN" for short) is a positive integer whose divisors add up to some larger multiple of the original number; that is, sigma(N) = kN for some integer k greater than 2. The first multiperfect number is 120; its divisors are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120, and they add up to 360 = 3*120. So for 120, k = 3; this is called the index of a multiperfect number.

Perfect numbers were known to the ancient Greeks; Euclid's works include a proof that if a number of the form (2^n - 1) is prime, then 2^(n-1) (2^n - 1) is perfect (e.g., n = 2 and n = 3 yield the perfect numbers 6 and 28). In the 18th century, Euler proved that any perfect number that was even must be of the form given by Euclid.

No one knows if any odd perfect numbers exist, but there are good reasons to doubt it. Euler's result means that searching for even perfect numbers is reduced to checking if 2^n - 1 is prime for different values of n. As of January 2000, 38 such Mersenne primes are known. The last four have been found by Internet users participating in the Great Internet Mersenne Prime Search (GIMPS).

Finding MPFNs is not so straightforward. They fit no known formula or pattern other than their definition. However, in recent years, several investigators have developed heuristic programs that have been very effective in discovering new MPFNs. Since early 1992, the number of known MPFNs has tripled from about 700 to well over 3000. Before 1992, no MPFNs were known with an index higher than 8; now more than 1300 are known with index 9 and 80 with index 10. No MPFNs with a higher index have yet been found.

In this space, I formerly asked for volunteers to help search for MPFNs. My sincere thanks to all those who expressed interest. In the past few years, my program has been greatly surpassed in the rate of MPFN discovery by Jason Moxham's; so if you want find MPFNs, visit his page. If that's not exactly how you'd like to put your computer's idle cycles to good use, try taking a look at the list of math-related distributed computing projects on the projects page.

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