What Is a Perfect Number? Numbers Equal to the Sum of Their Divisors

A perfect number equals the sum of its positive divisors excluding itself. Six is the first, since its divisors below itself are one, two and three, which sum to six. Twenty-eight is the second, from one, two, four, seven and fourteen. The next are four hundred and ninety-six and eight thousand one hundred and twenty-eight, and after that the numbers grow very quickly, with the fifth having eight digits. Numbers whose divisors sum to less than themselves are called deficient and those where the sum exceeds are abundant, and both are far more common. The Greeks knew the first four and attached significance to them, with the property treated as a mark of mathematical and sometimes mystical distinction, and later writers connected the first to the days of creation.

The structure of these numbers is understood and the understanding is ancient:

• •Euclid proved that a specific formula involving a power of two produces a perfect number whenever the accompanying factor is prime
• •Primes of the required form are one less than a power of two and are called Mersenne primes
• •Euler proved that every even perfect number has exactly that form, which completes the description for even numbers
• •Every known perfect number is therefore paired with a Mersenne prime, and finding one finds the other
• •Searching for such primes is the main activity, conducted by a large distributed computing project
• •The largest known has tens of millions of digits, which is why only a few dozen are known despite the effort

Two questions that can be stated in a sentence have resisted every attempt. Whether infinitely many perfect numbers exist is unknown, and it is equivalent to whether infinitely many Mersenne primes exist, which is a longstanding conjecture supported by heuristic arguments about how often such numbers should be prime and by the continued discovery of new ones. Whether any odd perfect number exists is unknown, and this is the more striking gap, since nobody has found one and nobody has proved none can exist. Extensive work has established a formidable list of properties any odd example must have, including being extremely large, having a minimum number of distinct prime factors and satisfying numerous other constraints, and searches have verified that none exists below an enormous bound. The constraints are so restrictive that most mathematicians expect none exists, which is not a proof.

The same idea of comparing a number with the sum of its divisors produces a small family of related definitions. Amicable pairs are two numbers each equal to the sum of the other's divisors, the smallest being a pair the Greeks knew and to which later writers attached significance. Sociable numbers extend that to longer cycles, where a chain of numbers each leads to the next and eventually returns to the start, and such chains of several members have been found. Semiperfect numbers equal the sum of some rather than all of their divisors. Weird numbers are abundant without being semiperfect, which is a surprisingly restrictive combination. Each of these has its own small literature and its own open questions, and together they illustrate how a single definition generates a family of problems by being varied slightly.

The practical value of these particular numbers is essentially nil and the reasons for studying them are worth stating honestly. They are a clean example of a problem statable to a schoolchild and unsolved after millennia, which is pedagogically valuable and demonstrates that elementary questions are not necessarily easy. The search for the associated primes has driven development in distributed computing and in algorithms for testing very large numbers for primality, and those algorithms have applications in cryptography where the subject matter does not. The constraints derived on hypothetical odd examples exercise techniques in number theory that transfer elsewhere. And the history connects ancient Greek mathematics directly to current computational work on the same question, which is unusual and is part of why the problem retains attention.

Six and twenty-eight equal the sums of their own divisors, and every even example has the form Euclid described, paired with a prime one less than a power of two. Only a few dozen are known because finding them means finding those primes. Whether infinitely many exist is unknown, and whether any odd one exists is unknown despite constraints so restrictive that most expect none.