Odd perfect numbers problem

[Kyriakos]

This is a yet unsolved- to my knowledge at least- problem in mathematics.

A perfect number is a number which is equal to the sum of all of its divisors, with those divisors giving a natural number as the result of their division with it, and excluding the number itself.

An example of such a number is 6, which is the smallest pefect number:

6= 1+2+3

The next one is 28 (1+2+4+7+14).


Relative theorems about the perfect numbers are centered on another group of numbers, the prime numbers (a prime is a number which is only divided by itself and 1 if its to give a result which is a natural number),

-Euclid (Ευκλείδη&#962 prooved that "If 2 raised in the power of k-1 is a prime number, then 2k-1(2k-1) is a perfect number and every even perfect number has this form"

-Euler (iirc) prooved that if 2 raised in the power of k-1 is a prime then k has to be a prime itself.

Also the "Mersenne primes" are relative to the attempt to find a proof of whether or not there exists an odd perfect number. http://mathworld.wolfram.com/MersennePrime.html



Since i am making progress with my literary work i am thinking of employing myself with a small novel where this issue is also part of the story. I am not yet decided how integral a part of it it shall be. If i get very into this problem i may even try to form a more in depth view of the contemporary syllogistic about it, which seems to be more targeted towards calculating facts about the possible odd perfect numbers.

Finally it has to be said that it has already been calculated that there is not one odd perfect number, with all numbers up to 10 raised to the power of 300, already checked

I am interested in hearing if anyone in the forums has also given thought to this problem. I do not recall any mathematician being here, only a couple of physicists, so possibly not?

 

[sanabas]

That formula is missing some signs. If (2^k)-1 is prime, then [(2^k)-1][2^(k-1)] is a perfect number.

 

[DBear]

I don't see how there can be an odd perfect number. By definition, an odd number does not have 2 as a factor, and therefore does not have half the number as a factor. There just aren't enough other factors to make up the difference. Of course this is no proof, but an observation.

 

[sanabas]

I don't see how there can be an odd perfect number. By definition, an odd number does not have 2 as a factor, and therefore does not have half the number as a factor. There just aren't enough other factors to make up the difference. Of course this is no proof, but an observation.

That one's very easy to shoot down. Abundant numbers have the sum of their factors greater than the number itself. So if not having half the number as a factor restricted the sum of the factors to less than the number itself, there would be no odd abundant numbers. There are an infinite number of odd abundant numbers, starting with 945. (1+3+315+5+189+7+135+9+105+15+63+21+45+27+35=975)

 

[Marshy]

Since i am making progress with my literary work i am thinking of employing myself with a small novel where this issue is also part of the story. I am not yet decided how integral a part of it it shall be. If i get very into this problem i may even try to form a more in depth view of the contemporary syllogistic about it, which seems to be more targeted towards calculating facts about the possible odd perfect numbers.

Determining properties that a certain object must have (in this case the objects are odd perfect numbers) is a good technique for showing that they don't exist. Every new property that the object must satisfy reduces the number of possible candidates. Eventually one hopes to reduce the number of candidates to 0.

To look at it another way if you show the object must have properties A and B and you can also show that nothing can simultaneously satisfy these two properties then you have shown that the object cannot exist.

 

[Souron]

10^300 is only 125 bytes in memmory. With a fast factoring algorithim it would not be very hard to write a program that tests much larger integers.

But I gues that's the hard part: a fast factoring algorithim for 300+ digit numbers.