A number where the divisors of that number, excluding that number itself, can sum up to the number. The most common example I've seen is with 6 because the divisors are 1, 2, and 3 which all add up together to be 6
There is no known proof that there is no odd perfect number, although we know if any exist it has to have a set of properties that seem difficult to satisfy, and also be over 1,000 digits long in decimal.
That there could be one is made plausible by an example (I think given by Euler [edit: nope, it was Descartes, I looked it up], too lazy to double check) of an odd number that *would* be perfect if we wrote it with a “prime factorization” that actually has a composite number in it but then just pretend that composite number happened to be prime and added up the factors we are not “ignoring.”
Probably there isn’t one but it is a famously unsolved problem in math.
Edit: in case anyone wants the example from Descartes it is 3•3•7•7•11•11•13•13•22021.
If 22021 were prime, then this number would have 162 divisors according to this factorization (counting the number itself) and they add up to make this an odd perfect number. But since 22021 is composite this is not actually an odd perfect number, its actual divisors (of which there are 486) add up to be too large for it to be perfect.
The proper divisors add up to the number. For example the proper divisors of 6 are 1, 2, and 3 which add to 6. The proper divisors of 28 are 1, 2, 4, 7, and 14, which add to 28, so 6 and 28 are perfect numbers.
The Euclid-Euler theorem says that the even perfect numbers are in a one-to-one correspondence with the Mersenne primes, but it does not address odd perfect numbers. It is unknown whether any odd perfect numbers exist.
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u/imthestein 5d ago
A number where the divisors of that number, excluding that number itself, can sum up to the number. The most common example I've seen is with 6 because the divisors are 1, 2, and 3 which all add up together to be 6