From: hook@nas.nasa.gov (Ed Hook) Subject: Re: 3-perfect even numbers Date: 29 Jul 2000 17:29:41 GMT Newsgroups: sci.math Summary: [missing] In article , Fred W. Helenius writes: |> Paul Bruckman wrote: :> >Hey, can anyone tell me what is known about even 3-perfect numbers? A :> >number n is 3-perfect iff sigma(n) = 3n , where sigma(n) is the sum of :> >divisors function. A visual search through a table yielded the :> >following two examples of 3-perfect numbers n <= 1000 : n = 120 = :> >2^3*4*5 and n = 672 = 2^5*3*7. |> Six 3-perfect numbers are known, and have been since the mid-17th |> century: |> 120 = 2^3 3 5 |> 672 = 2^5 3 7 |> 523776 = 2^9 3 11 31 |> 459818240 = 2^8 5 7 19 37 73 |> 1476304896 = 2^13 3 11 43 127 |> 51001180160 = 2^14 5 7 19 31 151 |> Although thousands of other multiperfect numbers have been (and |> continue to be) found, no other 3-perfects have emerged. It's known |> that there are no others up to 10^70, and it is suspected by many |> that there are no more to be found. (The useful multiperfect number |> page at http://www.uni-bielefeld.de/~achim/mpn.html goes so far as |> to assert that there are no more.) Unfortunately, this is beyond |> what we can prove today; in particular, it would constitute a proof |> that there are no odd perfect numbers, since twice an odd perfect |> would be 3-perfect. If n is an odd perfect number, then I'd calculate that sigma(2n) = sigma(2) sigma(n) = 3(2n) = 6n. Am I confused ? -- Ed Hook | Copula eam, se non posit Computer Sciences Corporation | acceptera jocularum. NAS, NASA Ames Research Center | All opinions herein expressed are Internet: hook@nas.nasa.gov | mine alone ============================================================================== From: hook@nas.nasa.gov (Ed Hook) Subject: Re: 3-perfect even numbers Date: 30 Jul 2000 15:52:03 GMT Newsgroups: sci.math In article , Deinst@world.std.com (David M Einstein) writes: |> Ed Hook (hook@nas.nasa.gov) wrote: |> : If n is an odd perfect number, |> : then I'd calculate that |> : sigma(2n) = sigma(2) sigma(n) |> : = 3(2n) |> : = 6n. |> : Am I confused ? |> Yes and no. 2*3 = 6, so 2n is triperfect. <* smacks forehead *> ... thanks ... *blush* -- Ed Hook | Copula eam, se non posit Computer Sciences Corporation | acceptera jocularum. NAS, NASA Ames Research Center | All opinions herein expressed are Internet: hook@nas.nasa.gov | mine alone ============================================================================== From: Fred W. Helenius Subject: Re: Perfect Numbers Date: Wed, 09 Aug 2000 10:16:20 -0400 Newsgroups: sci.math "Paulo J. Matos aka PDestroy" wrote: >Which numbers are perfect? By definition, a positive integer n is perfect if it is equal to the sum of its positive integer divisors, excluding itself. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14, so 6 and 28 are perfect. Euclid proved that if p is a prime number, and 2^p - 1 is also prime, then 2^(p-1)*(2^p - 1) is a perfect number. (6 and 28 result from setting p = 2 or 3.) Much later, Euler proved that there are no even perfect numbers other than those given by Euclid's formula. Of course, it remains to find those primes p for which 2^p - 1 is prime; this is a very difficult problem to which much computational effort has been devoted. See http://www.mersenne.org/prime.htm for recent progress. Whether there are any odd perfect numbers is a very old unsolved problem. Many severe restrictions are known on the possible form of an odd perfect number; so many that it seems doubtful that any exist. But it hasn't been proved one way or the other. For more information and references to related ideas, you might look at http://mathworld.wolfram.com/PerfectNumber.html . >Why? Why? Because that's the definition of "perfect number". If you're asking why these numbers were singled out, it goes back to ancient Greek numerology; they remain of interest because they relate to important parts of number theory. -- Fred W. Helenius