Are there inf. many perfect numbers?
James Ųyvind Baum
Are there infinitely many perfect numbers?
Does anyone have a list of perfect numbers? eg. 6 28 496 8128
Please mail these to me at
thanks
james
Chance Harris
: Are there infinitely many perfect numbers?
: Does anyone have a list of perfect numbers? eg. 6 28 496 8128
yes, for any prime p:
(2^p -1 ) * (2 ^ (p-1))
is perfect.
I don't know if this is garuanteed to generate all of them.
Chance
#include<std_disclaimer.h>
Keith Ramsay
| yes, for any prime p:
|
| (2^p -1 ) * (2 ^ (p-1))
| is perfect.
No, for example, when p=11 this is not perfect.
The even perfect numbers are exactly those numbers
of the form (2^p-1)*(2^(p-1)), where 2^p-1 is prime.
This only can happen when p is prime. The primes of
the form 2^p-1 are known as Mersenne primes. We do not
know whether there are infinitely many of them, but it
seems likely that there are.
We don't know whether there are any odd perfect numbers.
Probably there aren't.
Keith Ramsay
ram...@math.ubc.ca
Edward C. Hook
chance@wetware (Chance Harris) writes:
|> James Ųyvind Baum (jam...@ifi.uio.no) wrote:
|>
|> : Are there infinitely many perfect numbers?
|> : Does anyone have a list of perfect numbers? eg. 6 28 496 8128
|>
|> yes, for any prime p:
|>
|> (2^p -1 ) * (2 ^ (p-1))
|> is perfect.
Bzzzt! Thank you for playing ... _If_ 2^p - 1 is prime, then your expression
gives a perfect number, and all _even_ perfect numbers arise in this way.
Whether there are _any_ odd perfect numbers has been an open question since
the time of the ancient Greeks. Since it is also unknown whether there are
an infinitude of Mersenne primes ( == primes of the form 2^p - 1 ), the
answers to 'jamesb's questions are:
(a) We don't know.
(b) Check out the entry for "What is the current status on Mersenne
primes?" under the 2nd question in Part I of the sci.math FAQ list;
from the list given there, you will be able to generate the numbers
you want (with the application of sufficient patience).
|>
|> I don't know if this is garuanteed to generate all of them.
|>
|> Chance
|> #include<std_disclaimer.h>
|>
|>
|>
|>
|>
|>
|>
--
Ed Hook | Coppula eam, se non posit
Computer Sciences Corporation / NAS | acceptera jocularum.
NASA/Ames Research Center | Me? Speak for my employer?...<*snort*>
Internet: ho...@nas.nasa.gov | ... Get a _clue_ !!! ...
Benjamin V.C. Collins
>James Ųyvind Baum (jam...@ifi.uio.no) wrote:
>: Are there infinitely many perfect numbers?
>: Does anyone have a list of perfect numbers? eg. 6 28 496 8128
>yes, for any prime p:
>(2^p -1 ) * (2 ^ (p-1))
>is perfect.
I might have fallen asleep during my number theory class, but
doesn't this break down at about p=11?
--BVCC
--
Benjamin V.C. Collins * "Because of circumstances beyond my
col...@math.wisc.edu * control, I am captain of my fate
* and master of my soul." --Ashleigh Brilliant
Chance Harris
: Chance Harris writes
: Keith Ramsay
: ram...@math.ubc.ca
woops
Chance
Chris Thompson
|>
|> Bzzzt! Thank you for playing ... _If_ 2^p - 1 is prime, then your expression
|> gives a perfect number, and all _even_ perfect numbers arise in this way.
|> Whether there are _any_ odd perfect numbers has been an open question since
|> the time of the ancient Greeks.
This business about the "ancient Greeks" is often stated in this newsgroup when
perfect numbers come up, but without references. This time I thought I would
get out my sources (secondary: mostly Heath). >Informed< corrections welcomed...
It was Euclid (c. 300 BC), in Elements Book VII Definition 22, who first used
the term "perfect number" (teleios arithmos) in the modern sense. (He seems to
have borrowed the phrase from the Pythagoreans who used it for something less
mathematically interesting.) The one and only thing he proves about them is in
Elements Book VIII Proposition 36, that if 2^n-1 is prime then 2^(n-1)*(2^n-1)
is perfect, or in the original (Heath's translation):
If as many numbers as we please beginning from an unit be set out
continuously in double proportion, until the sum of all becomes prime,
and if the sum multiplied into the last makes some number, the product
will be perfect.
Euclid arranges this as the climax of the "Number Theory" section of the
Elements, i.e. books VI to VIII.
Now, having proved this, it seems a pretty obvious question to ask "are
all perfect numbers of this form?". But I can find no reference to any of
the "ancient Greeks" actually posing this question explicitly. Some later
authors seem to have just assumed that this was the case.
In fact those later surviving works which do mention perfect numbers seem
to have been "Fun with Figures" level stuff, rather than serious mathematics,
and their authors are hardly worried about things like >proofs<. (They would
been right at home in sci.math.) They are:
Nicomachus (c. 100 AD) "Introduction to Arithmetic" Book 1 Chapter 16.
He is the first to describe what we now call "abundant" and "deficient"
numbers. He gives the first four perfect numbers (6, 28, 496, 8128) and
suggests that they end alternately in 6 and 8 [wrong] and that there is
one in each interval between successive powers of 10 [wrong].
Theon of Smyrna (c. 130 AD) "Exposition on mathematical matters useful for
reading Plato" (an excuse for a long ramble around mathematics of very
little relevance to anything in Plato).
Covers the same sort of stuff as Nicomachus.
Iamblichus (c. 300-350 AD) Commentary on Nicomachus.
He amends the "successive powers of 10" by suggesting that it may change
over to being "successive powers of 10000" [still wrong]. (The myriad, 10000,
of course, being the canonical big-power-of-10 for the Greeks, in the same way
as 1000 is for us). Heath suggests that this means that he knew that the
fifth perfect number was 33550336, but it isn't given explicitly.
Iamblichus is the first to describe "amicable pairs" of numbers, although
he attributes the concept to Pythagoras himself.
Chris Thompson
Internet: ce...@phx.cam.ac.uk
JANET: ce...@uk.ac.cam.phx
Chris Thompson
|> [...] >Informed< corrections welcomed...
and then managed to misnumber the books of Euclid's Elements
|> Elements Book VIII Proposition 36, that if 2^n-1 is prime then 2^(n-1)*(2^n-1)
^^^^
should be Book IX
|> Euclid arranges this as the climax of the "Number Theory" section of the
|> Elements, i.e. books VI to VIII.
^^^^^^^^^^
should be books VII to IX.
Apologies.
Bill Taylor
|> Now, having proved this, it seems a pretty obvious question to ask "are
|> all perfect numbers of this form?". But I can find no reference to any of
|> the "ancient Greeks" actually posing this question explicitly. Some later
|> authors seem to have just assumed that this was the case.
Certainly the question of whether or not "are there any odd perfect numbers"
holds the undisputed folklore championship of oldest unsolved math problem;
by far. When this topic last arose, the only outright statement I could
find a reference for was...
"The Man who loves only numbers" Paul Hoffman
Atlantic Monthly, Nov '87 , page 87
where he states
"the Greeks wondered whether odd perfect numbers exist" and that it is
"the oldest unsolved problem in maths".
Of course, Hoffman might still merely be one of the "later authors" who just
assumed so. Unfortunately I can't recall now exactly what he said or what
sources he gave, (probably none). You'd have to check the article.
Meanwhile, assuming that it *is* the oldest unsolved problem, let me ask again:
what is the best candidate for second-oldest unsolved math problem ?
^^^^^^^^^^^^^
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Bill Taylor w...@math.canterbury.ac.nz
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Q: What happened when the creationist finally changed his mind?
A: It didn't get any better.
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