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
5 thoughts on “Odd perfect numbers”
A short abstracts outline the strategy. I didn’t read the details.
I (humbly) invite you to take a look at http://arnienumbers.blogspot.com.
It contains (among other things) an attempt to prove the OPN Conjecture via a “Tour de Force”.
Do let me know if you have any questions!
The problem is, we only know that what our current boundry of numbers encompasses. 10To the 300th has been the extent of the search for odd perfect numbers. If you expand your search beyond the scope of current computer and mathematical capabilities, then maybe an odd perfect number does exist. Our current knowledge is so limited to what a computers can generate solutions, that maybe we should realize we are at the very smallest part of infinty. Mathematical tactics and theory can only cover a small part of the entire number set. This, of course, is a logical arguement, not a mathematical arguement.
I just rediscovered this post, and a little bit of searching turn up a newer, greater, bound.
ODD PERFECT NUMBERS ARE GREATER THAN 101500
PASCAL OCHEM AND MICHAEL RAO ¨
Abstract. Brent, Cohen, and te Riele proved in 1991 that an odd perfect
number N is greater than 10^300. We modify their method to obtain N >
10^1500. We also obtain that N has at least 101 not necessarily distinct prime factors and that its largest component (i.e. divisor p
a with p prime) is greater than 10^62.
Ochem, P., & Rao, M. (2012). Odd perfect numbers are greater than 10¹⁵⁰⁰. Mathematics of Computation, 81(279), 1869-1877.
Still more searching give this, which I can’t access: https://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02941-X/
^^^ found it, but it’s a _deep_ read.