The Greeks called a number perfect when it was exactly the sum of its smaller divisors: 6 is 1 + 2 + 3, and 28 is 1 + 2 + 4 + 7 + 14. They are rare. After more than two thousand years, fifty-two are known, the largest over eighty million digits long, and every one of them is even, each built from a prime of a particular kind, as Euclid half-showed and Euler finished. Nobody has ever found an odd one, and nobody has proved there isn’t one. What the searching has produced instead is a list of conditions: an odd perfect number, if there is one, is larger than 10^1500, has at least ten different prime factors, and the biggest of them is over a hundred million. It is an odd sort of progress, a description getting steadily more precise of something no one has seen, and of how large and complicated a thing would have to be to manage being both odd and perfect.