OddPerfectNumbers

Although some even PerfectNumbers exist, it is unknown if there exist odd PerfectNumbers.

What are we searching for?

An Odd Perfect Number. That is, some integer equal to the sum of its proper divisors (i.e. all divisors except itself).

Example: 6 is an even perfect number, because 1 + 2 + 3 = 6.

`How many are there?`



None.... that we know of yet.  It has been conjectured that they do not exist, but that is unproven.



  What kind of bounds have been placed on odd PerfectNumbers? 



 Numbers up to 10³⁰⁰ have been checked [8]





 What's currently being done? 


 ??? -- don't know





  What kinds of things do we know about OddPerfectNumbers that may be of use? 


 "The sum of reciprocals of all the divisors of a perfect number is 2" [1]

 "Euler showed that an odd perfect number, if it exists, must be of the form m = p^4r+1Q², where p is a prime of the form 4n + 1" [2]




 "if one [odd perfect number] exists, a lot is known about it:

 it is a perfect square multiplied by an odd power of a single prime

 it has at least 8 distinct prime factors

 it has at least 29 prime factors (not necessarily distinct)

 it has a prime factor greater than 10⁶

 its second largest prime factor is greater than 10⁴

 its third largest prime factor is greater than 10²

 it is divisible by a prime component greater that 10²⁰"  [3]






 "an odd perfect number would have to have at least eight distinct prime factors, and at least 29 prime factors which are not necessarily distinct. It is also known that such a number would have more than 300 digits and a prime divisor greater than 10⁶." [6]






 Bibliography 

 [1] http://mathworld.wolfram.com/PerfectNumber.html

 [2] http://mathworld.wolfram.com/OddPerfectNumber.html

 [3] http://www-users.cs.york.ac.uk/~susan/cyc/p/perfect.htm




 [4] http://www.primepuzzles.net/conjectures/conj_011.htm  -- this is listed as the 11th 'prime puzzle' !

 [5] http://www-maths.swan.ac.uk/pgrads/bb/project/node13.html

 [6] http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html -- nice, understandable article.

 [7] http://primes.utm.edu/glossary/page.php?sort=PerfectNumber

 [8] http://web.comlab.ox.ac.uk/oucl/work/richard.brent/pub/pub116.html -- the homepage for Brent (et al.)'s paper proving a lower bound of 10³⁰⁰ on odd perfect numbers.







RobinsonTryon - 14 Dec 2002