Date: Wed, 16 Dec 1998 14:35:17 -0600 (CST)
To: Luxemburger@snafu.de
Subject: Re: how to find prefect numbers ?
You wrote:
>Please can anyone help me to find a good method to find "vollkommene
>Zahlen"(or Perfect Numbers)
>
>an example is 6 -> is divided by 1 +2+3
>the sum of these divisors is 6 again !
No odd perfect numbers are known. It has not been proven that none exist,
but results have been proven which would require such a number to have
many prime factors to high exponents, so it is generally assumed that none
exist. (I believe that the question of whether odd perfect numbers exist
is the oldest open question in mathematics, now about 2 and a half millenia
old!)
It is known that an even number is perfect iff it is of the form
2^(p-1) * ((2^p) - 1)
where the second factor is prime. Primes of the form 2^p-1 are known as
Mersenne primes. It is known that this expression can only be prime if p
itself is prime, and there are fairly efficient ways to determine whether a
number 2^p - 1 is prime, so very large primen numbers, and thus very
large perfect numbers, have been found in this way. Indeed, what is usually
true and is true today is that the largest number known to be prime is a
Mersenne prime. About 3 dozen are known, with up to a million digits
(giving a two-million digit perfect number).
For more information, consult any elementary number theory book. You
might be interested in Ribenboim's Book of Prime Number Records, or Guy's
Unsolved Problems in Number Theory, for example.
More information about testing Mersenne numbers for primality is at
http://www.math.niu.edu/~rusin/known-math/index/11Y05.html
and its parent page.
[deletia -- djr]
==============================================================================
Subject: Perfect numbers (from Mathematical Reviews on the Web)
From: rusin
Date: Dec 18 1998 08:52 (datestamp)
To: get current bounds on odd-perfect-number searches :-)
Selected Matches for: Anywhere=(odd perfect number)
98k:11002 11A25 (11Y70)
Hagis, Peter, Jr.(1-TMPL); Cohen, Graeme L.(5-UTSY)
Every odd perfect number has a prime factor which exceeds $10\sp 6$.
(English. English summary)
Math. Comp. 67 (1998), no. 223, 1323--1330. [ORIGINAL ARTICLE]
The authors prove that if $N$ is an odd perfect number then $N$ has a
prime factor which exceeds $10\sp 6$. This result is an improvement of
all known results in this subject, such as, among others, the result
of M. S. Brandstein announced in [Abstracts Amer. Math. Soc. 3 (1982),
no. 1, 257, Abstract 82T-10-240] that at least one of the primes such
that $p\sb i\mid N$ is greater than $5·10\sp 5$. In the proof of this
result the authors use some arithmetical properties of the cyclotomic
polynomial and several interesting lemmas and propositions whose
proofs are given by theoretical considerations and computer
calculations.
Reviewed by Aleksander Grytczuk
_________________________________________________________________
96f:11009 11A25 (11Y70)
Cook, Roger(4-SHEF-SM)
Factors of odd perfect numbers. (English. English summary) Number
theory (Halifax, NS, 1994), 123--131,
CMS Conf. Proc., 15,
Amer. Math. Soc., Providence, RI, 1995.
If $N$ is odd, let $p\sb 110\sp {30}$. The proof
requires a great deal of computation, which is not carried out in the
paper but can be obtained from the author. He also obtains other
inequalities for odd perfect numbers.
Reviewed by P. Erdos
_________________________________________________________________
86f:11010 11A25
Wagon, Stan(1-SMTH)
The evidence: perfect numbers.
Math. Intelligencer 7 (1985), no. 2, 66--68.
The author briefly surveys the history of the problem of the
nonexistence of odd perfect numbers.
_________________________________________________________________
86f:11009 11A25
Cohen, G. L.(5-NSWIT); Williams, R. J.(5-NSWIT)
Extensions of some results concerning odd perfect numbers.
Fibonacci Quart. 23 (1985), no. 1, 70--76.
As is well known, an odd perfect number $N$, if one exists, must be of
the form $N=p\sp \alpha q\sp {2\beta\sb 1}\sb 1 q\sp {2\beta\sb 2}\sb
2\cdots q\sp {2\beta\sb \tau}\sb t$ for distinct odd primes $p,q\sb
1,q\sb 2,\cdots,q\sb t$, with $p\equiv\alpha\equiv1\bmod4$. The
authors add the following to previously known results: If $\beta\sb
1=\beta\sb 2=\cdots=\beta\sb t=\beta$, then $\beta\ne6,8,11,14$ or 18;
and if $\beta\sb 2=\cdots=\beta\sb t=1$, then $\beta\sb 1\ne5$ or 6.
They also show that if $x$ is the number of prime powers $q\sp
{2\beta\sb i}\sb i$ for which both $q\sb i\equiv1\bmod 4$ and
$\beta\sb i\equiv1\bmod2$, then $p-\alpha\equiv4x\bmod8$.
Reviewed by V. C. Harris
© Copyright American Mathematical Society 1998