Here are three of several classic unsolved problems in mathematics. The problem statements are listed first, with definitions and background, as appropriate, below.
Mystery #1: Are the pairs of twin primes infinite? (Do they go on forever or end at some point?)
Mystery #2: Are the perfect numbers infinite? (Do they go on forever . . .?)
Mystery #3: Are there any odd perfect numbers?
BACKGROUND:
Mystery #1: Primes are positive integers (other than 1) that are divisible only by themselves and 1. Twin Primes are primes that are only two apart: 3 & 5, 17 & 19, 101 & 103, etc. Interestingly, it has been known for a long time that the primes themselves are infinite. Euclid proved this roughly 200 years before Christ. So, this is one of those cases where it seems like it should be easy to settle, but no one has been able to prove that they go on forever or prove that they eventually quit.
Mystery #2: Perfect numbers are those whose factors (other than the number itself) add up to the original number. Examples: The factors of 6 (other than 6 itself) are 1,2, & 3. 1 + 2 + 3 = 6, so 6 is perfect. 28 is also perfect, since 1 + 2 + 4 + 7 + 14 = 28. The next two are 496 and 8128, and these four were known to the ancient Greeks. There are currently only 48 known perfect numbers – and several of them are HUGE, but no one has succeeded in proving they go on forever or that they eventually stop.
Side Note: Perfect numbers have a surprising and fascinating connection to Mersenne Primes (mentioned in earlier Math Tidbits), but this will have to wait for another Math Tidbit sometime.
If a perfect number is expressed by the formula below, then all perfect numbers are divisible by 2 which automatically means they are all are even. [From Wikipedia – merrsenne prime exponent p which generates them with the expression 2p−1× (2p − 1) where 2p − 1]
However, apparently a Mersenne Prime is not necessary to find perfect numbers if there is still a possibility (no matter how small) that there is an odd perfect number.
So, how is the search for perfect numbers conducted?
Excellent observations, Jim. A Mersenne prime is not ‘necessary’, but . . . . Here is the precise situation, trying to keep it as non-technical as possible:
1. It can be proven that for EVERY Mersenne prime (no one knows if they’re infinite either), there is an associated perfect number, which, as you point out, has to be even. So there are currently 48 (or is it 47?) known Mersenne primes and 48 (47?) known (even) perfect numbers.
2. AND, it can be proven that if you START with an EVEN (note that important condition!!) perfect number, it MUST be of the form above that you refer to. SO, Mersenne Primes and perfect numbers are forever inextricably linked. (One of the most amazing coincidences in all of mathematics, in my book!! And also the marvelous relationship to which I referred.)
3. But it is still true that no one KNOWS (one way of the other) if there are any ODD perfect numbers. I’m tempted to say that the odds are overwhelming that there aren’t any, but that would be poor mathematics. 🙂 No one can find one [and they have checked up into the HUGE mega-big numbers], but so far no one can prove there can’t be any on out there even further.
Jim –
I notice I didn’t completely answer your question. As to how the search for perfect numbers is conducted:
In some ways, one doesn’t need to actually search for (even) perfect numbers. The search is for Mersenne primes, which is ‘easier’ (definitely a relative term!!). Whenever a new Mersenne prime is found, then – given the connection mentioned earlier – then there is automatically a new (even) perfect number that comes along for free.
[Side note: The search for Mersenne primes is SO organized, that you (your computer) could actually join the search. Google GIMPS: Great Internet Mersenne Prime Search, and sign on to put your computer to work. Perhaps the 49th MP will be discovered by Jim Waterman!! In fact, THREE of the 48 known MPs were discovered by our colleagues at University of Central MO at Warrensburg!]
Now, for odd perfect numbers, the situation is slightly different. One is stuck with EITHER hoping one exists and hunting for it – perhaps forever – OR hoping one doesn’t and then trying to PROVE that fact – clearly elusive and difficult. Or, of course, BOTH. It is known that IF an odd perfect number exists, it must be BIG (you could probably google odd perfect numbers and discover the lower bound) – another relative term!
Don’t know if that answers your question or not.