Yesterday I wrote about even perfect numbers. What about odd perfect numbers? Well, there may not be any.
I couldn’t care less about perfect numbers, even or odd. But I find the history and the mathematics surrounding the study of perfect numbers interesting.
As soon as you define perfect numbers and start looking for examples, you soon realize that all your examples are even. So people have wondered about the existence of odd perfect numbers for at least 2300 years.
No one has proved that odd perfect numbers do or do not exist. But people have proved properties that odd perfect number must have, if there are any. So far, although the requirements for odd perfect numbers have become more demanding, they are not contradictory and it remains logically possible that such numbers exist. However, most experts believe odd perfect numbers probably don’t exist. (Either odd perfect numbers exist or they don’t. How can one say they “probably” don’t exist? See an explanation here.)
Wikipedia lists properties that odd perfect numbers must have. For example, an odd perfect number must have at least 300 digits. It’s interesting to think how someone determined that. In principle, you could just start at 1 and test odd numbers to see whether they’re perfect. But in practice, you just won’t get very far.
A year is about 10^7.5 seconds (see here). If you had started testing a billion (10^9) numbers a second since the time of Euclid (roughly 10^3.5 years ago) you could have tested about 10^20 numbers by now. Clearly whoever came up with the requirement N > 10^300 didn’t simply use brute force. There may have been some computer calculation involved, but if so it had a sophisticated starting point.
Related: Applied number theory