From: gerry@mpce.mq.edu.au (Gerry Myerson)
Subject: Re: 2n = the sum of its divisors
Date: Thu, 11 Mar 1999 15:33:08 +1100
Newsgroups: sci.math
Keywords: multiply-perfect numbers
In article , "Dominic"
wrote:
=> while doing a little exercise on perfect numbers, i was running a comparison
=> of numbers to their sum of factors (including 1 and excluding the number
=> itself) the only 3 numbers i was able to find that were equal to half their
=> sum of divisors were 120, 672, and 523776 (i'd appreciate it if someone
=> could check my math on that last one) are there an infinite number of these
=> things? anyone have a suggestion for generalizing them?
These are called multiply-perfect numbers.
See Problem B2 of Guy, Unsolved Problems in Number Theory.
==============================================================================
From: fredh@ix.netcom.com (Fred W. Helenius)
Subject: Re: 2n = the sum of its divisors
Date: Thu, 11 Mar 1999 05:17:11 GMT
Newsgroups: sci.math
"Dominic" wrote:
>while doing a little exercise on perfect numbers, i was running a comparison
>of numbers to their sum of factors (including 1 and excluding the number
>itself)
>the only 3 numbers i was able to find that were equal to half their sum of
>divisors were 120, 672, and 523776 (i'd appreciate it if someone could
>check my math on that last one)
>are there an infinite number of these things?
Only six are known; and no new ones have been found for centuries.
But it's beyond current mathematics to prove that there are no more;
in particular, it would include solving the ancient problem of whether
odd perfect numbers exist (since twice an odd perfect would be one of
these "triperfect" numbers).
>anyone have a suggestion for generalizing them?
Let the sum of divisors be any integer multiple of n; then you have
what are called "multiperfect" numbers. For example, 30240 and 32760.
There may be infinitely many multiperfects; so far, more than 3000 are
known. See http://www.uni-bielefeld.de/~achim/mpn.html for a nearly
up-to-date list and historical information.
--
Fred W. Helenius
==============================================================================
From: fredh@ix.netcom.com (Fred W. Helenius)
Subject: Re: If Im not mistaken (perfect numbers)
Date: Sun, 18 Jul 1999 12:00:45 GMT
Newsgroups: sci.math
Deinst@world.std.com (David M Einstein) wrote:
> On a not entirely unrelated note, I have seen mention of the
>fact that all the triperfect numbers are known. How are sufficient
>bounds proved to know that all have been found?
This has not been proved, nor is it likely to be in the near future.
In particular, showing that there are no more triperfect numbers
than the six currently known would also prove that there are no odd
perfect numbers, since twice an odd perfect number is triperfect.
It is the case that some of those investigating multiperfect numbers
feel quite confident that all of index 3, 4, 5 and 6 have been found,
but there are no proofs to back these assertions.
Up-to-date information on what multiperfect numbers are currently
known is at http://www.uni-bielefeld.de/~achim/mpn.html (or was,
the page seems to have gone missing this morning).
--
Fred W. Helenius