Odd Perfect Numbers

The fifth perfect number broke the textbook. 33,550,336 has eight digits. Nicomachus of Gerasa, a moralizing, Neopythagorean arithmetician, had told readers the fifth would have five, and that perfect numbers would end in 6 and 8, turn by turn. Ismail ibn Fallus, listing perfect numbers in the thirteenth century, already had 33,550,336 on the page. The sixth even perfect number ends in 6, the same final digit as the fifth. The schedule was a wish.

The thing he was scoring is simple. Take the proper divisors, every positive divisor except the number itself. Six gives one, two, and three, and those add back to six. Twenty-eight gives one, two, four, seven, and fourteen. A short sum is deficient. A long sum is abundant. An exact sum is perfect, and Nicomachus talked about that exactness as a kind of virtue. The four perfect numbers he could name were 6, 28, 496, and 8128. All four are even. He declared that every perfect number is even. That declaration is still open.

Euclid, centuries before Nicomachus, already had a machine for even ones. When 2^p - 1 is prime, multiply it by 2^(p-1). The product is perfect. Six is 2 times 3. Twenty-eight is 4 times 7. Four hundred ninety-six is 16 times 31. A power of two sits in front of every output, so the machine produces even numbers only.

On 15 November 1638 René Descartes wrote to Marin Mersenne, still hunting an odd one. He offered 198,585,576,189, built from 3 squared, 7 squared, 11 squared, 13 squared, and 22021. Treat 22021 as prime and the proper divisors add to the number. Factor 22021 as 19 squared times 61 and the equality fails. Descartes had found a spoof. He still believed a real odd perfect number could be hunted down. The hunt outlasted the letter.

Leonhard Euler closed the even side in a paper read to the Berlin Academy on 23 February 1747. Every even perfect number comes from Euclid’s machine, one for each prime of the form 2^p - 1. He also locked the shape of any odd perfect number. It must be a prime q raised to a power α, times a square that q does not divide, with both q and α leaving remainder 1 when divided by 4. The conditions have kept tightening: a gigantic lower bound, a crowd of distinct prime factors, and still no example. Euler called the existence question a most difficult one.

The even side has a finished description and a growing list of examples. Each new Mersenne prime, a prime of the form 2^p - 1, hands over one new even perfect number by Euclid’s rule. Fifty-two such primes are known, so fifty-two even perfect numbers are known. Whether further Mersenne primes keep appearing forever is its own open question. That question does not touch odd integers.

Any odd perfect number, if one exists, has to wear Euler’s shape: q^α m^2, with q prime, q not dividing m, and both q and α congruent to 1 modulo 4. On top of that shape, published constraints force the number to be enormous. Pascal Ochem and Michaël Rao showed in 2012 that it exceeds 10^1500 and has at least 101 prime factors counting multiplicity. Pace Nielsen showed in 2015 that it has at least 10 distinct prime factors. The largest of those primes exceeds 10^8. And 105 cannot divide it.

Carl Pomerance has given a heuristic that expects the odd column to stay empty. A heuristic estimates how often the required pieces should line up. The estimate is thin enough that an example would be a surprise. The logical status is the same either way. No odd perfect number has been found, and no proof rules them out.