It's astonishing how little we know about multiply-perfect numbers.. Reminder: let sigma(n) be the sum of all divisors of n, including 1 and n. A number n is multiply-perfect (of order k) if sigma(n) = kn. The classical perfect numbers are exactly the multiply-perfect numbers of order 2. We do not know if there are any odd multiply-perfect numbers, other than 1. Let me restate this in a slightly weird way: we do not know if there are any multiply-perfect numbers that are multiples of 1 but not of 2, outside the set {1} But our ignorance is much, much more profound than that. To my knowledge, we can't prove any theorem of the form: "There are no multiply-perfect numbers that are multiples of N but not of M, outside the set S." In particular, for example, we do not know if there are any multiply-perfect numbers (other than 1) whose prime factors are all larger than 100. It seems staggeringly unlikely that there would be any, and it is easy to put astronomical lower bounds on such numbers, but we still have no proofs. I'm quite sure this was all true about 30 years ago. Is it still true?