Any news on odd perfect numbers?
Mark Griffith
Ciao, Mark
KRamsay
con...@otherlanguages.org (Mark Griffith) writes:
>Anyone know of any recent work on the odd perfect number question?
I remember seeing some lower limits (like on the number of prime divisors
or something of the kind) being pushed up sometime in the past few years.
Keith Ramsay
Dan Hoey
> Anyone know of any recent work on the odd perfect number
> question?
Oddly enough, a paper claiming to prove the conjecture has
appeared, dated Thu, 8 Jan 2004, at
http://www.arxiv.org/abs/hep-th/0401052/
I haven't got the background to tell whether it's good or bad.
Dik T. Winter
If I correctly read what is written in that paper, the first error is
around the top of the first page. I do not know whether 1.1 can be
satisfied.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
John Ramsden
It doesn't look very promising to me, although admittedly I've only
skimmed through it.
For a start, one of the subject classifications 'High Energy Physics'
isn't relevant, or at least I didn't see in the paper any explanation
of why it might be, and that doesn't inspire much confidence at the
outset.
(It could be he feels obliged to add it because his institution or
department specializes in physics. Seems odd, but I guess that would
account for it.)
Also, in the paper itself, despite being reasonably formatted and
having some sensible looking references, the proof itself is at the
very least abominably badly presented.
He launches straight into in intricate series of manipulations with
no prior introduction or summary, and no explanation of what 'k' is
in the first equation, for example.
That in itself wouldn't be enough to condemn it out of hand; but just
at a first glance there are even more worrying features. For example,
throughout the proof, fractions pop up which it appears he may be
assuming without justification are integers (unless that is taken
care of in one of the references, but if so why not mention it?)
It may all be impeccably correct; but for what it's worth I'd bet
a fair amount that the whole thing is nonsense.
Cheers
John Ramsden
Robin Chapman
> hao...@aol.com (Dan Hoey) wrote in message
> news:<20040114233049...@mb-m22.aol.com>...
>> On Fri, 9 Jan 2004, Mark Griffith asked:
>> > Anyone know of any recent work on the odd perfect number
>> > question?
>>
>> Oddly enough, a paper claiming to prove the conjecture has
>> appeared, dated Thu, 8 Jan 2004, at
>> http://www.arxiv.org/abs/hep-th/0401052/
>> I haven't got the background to tell whether it's good or bad.
>
> It doesn't look very promising to me, although admittedly I've only
> skimmed through it.
>
> For a start, one of the subject classifications 'High Energy Physics'
> isn't relevant, or at least I didn't see in the paper any explanation
> of why it might be, and that doesn't inspire much confidence at the
> outset.
>
> (It could be he feels obliged to add it because his institution or
> department specializes in physics. Seems odd, but I guess that would
> account for it.)
It would also account for the loathsome practice
(common amongst fizzisists) of omitting titles of papers
in references [9, 12].
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Needless to say, I had the last laugh."
Alan Partridge, _Bouncing Back_ (14 times)
Phil Carmody
> In article <20040114233049...@mb-m22.aol.com> hao...@aol.com (Dan Hoey) writes:
> > On Fri, 9 Jan 2004, Mark Griffith asked:
> > > Anyone know of any recent work on the odd perfect number
> > > question?
> >
> > Oddly enough, a paper claiming to prove the conjecture has
> > appeared, dated Thu, 8 Jan 2004, at
> > http://www.arxiv.org/abs/hep-th/0401052/
> > I haven't got the background to tell whether it's good or bad.
>
> If I correctly read what is written in that paper, the first error is
> around the top of the first page. I do not know whether 1.1 can be
> satisfied.
What about
A_i = (q_i-1)((4k+1)^(4m+2)-1)
B_i = (4k) (q_i^(2^alpha_i+1)-1)
g=gcd(A_i,B_i)
a_i=A_i/g
b_i=B_i/g
?
However, I'm not putting any money on 1.2 being meaingful or not.
hep-th = High Energy Physics, Theory,
Hmmm... I thought General Math was the kooks' hangout?
I'm glad I normally surround myself with mathematical matters, as this
physics stuff, particularly the stuff Simon Davis is into, looks
pretty heavy:
http://www.iop.org/EJ/S/UNREG/G6Lj4JvumVUVaKYOUtyVpw/abstract/-search=5223505.2/0264-9381/18/17/305
<<<
Abstract. String propagation in ten-dimensional Minkowski space or the
direct product of Minkowski space and a six-dimensional Kähler
manifold or orbifold might be regarded as an approximation to a theory
which allows for the local curvature of spacetime by the
energy-momenta of the component fields. String scattering in the
interaction region might then be based on quantum field theory in a
local region with a curved geometry. Special emphasis is given to
field theory in anti-de Sitter space, as it represents a maximally
symmetric spacetime of constant curvature which could arise in the
description of matter interactions in local regions of
spacetime. Curvature shifts in the momentum and squared mass are
evaluated for scalar fields in anti-de Sitter space, and it is shown
that the shift in p2 + m2 compensates the ground-state contribution to
the bosonic string Hamiltonian, implying the consistency of computing
the scattering entirely in flat space. Dual space rules for evaluating
Feynman diagrams in Euclidean anti-de Sitter space are initially
defined using eigenfunctions based on generalized plane waves. Loop
integrals can be evaluated even more easily using momentum space rules
in conformally flat coordinates for anti-de Sitter space, which admits
flat three-dimensional sections that are analytic continuations of
horospheres in hyperbolic space H4. An additional argument in favour
of the model of string propagation described in this paper is based on
the removal of reflective boundary conditions on quantum fields
interacting in a locally anti-de Sitter region without spatial
infinity, implying the existence of a one-parameter family of
O(3,2)-invariant vacua in this region consistent with the degree of
freedom in defining the string theory vacuum.
>>>
http://www.iop.org/EJ/abstract/0264-9381/11/5/007
<<<
Abstract. The divergences that arise in the regularized partition
function for closed bosonic string theory in flat space lead to three
types of perturbation series expansions, distinguished by their genus
dependence. This classification of infinities can be traced to
geometrical characteristics of the string worldsheet. Some categories
of divergences may be eliminated in string theories formulated on
compact curved manifolds.
>>>
http://www.iop.org/EJ/S/UNREG/G6Lj4JvumVUVaKYOUtyVpw/abstract/-search=5223505.1/0264-9381/20/13/331
<<<
Abstract. The quantum cosmological wavefunction for a quadratic
gravity theory derived from the heterotic string effective action is
obtained near the inflationary epoch and during the initial Planck
era. Neglecting derivatives with respect to the scalar field, the
wavefunction would satisfy a third- order differential equation near
the inflationary epoch which has a solution that is singular in the
scale factor limit a(t) -> 0. When scalar field derivatives are
included, a sixth-order differential equation is obtained for the
wavefunction and the solution by Mellin transform is regular in the a
0 limit. It follows that inclusion of the scalar field in the
quadratic gravity action is necessary for consistency of the quantum
cosmology of the theory at very early times.
>>>
He seems in a very specialist field - "quantum cosmological wavefunction"
yields only 19 hits in google, of which 18 pertain to Simon Davis' own
papers.
And what is the "research foundation of southern california"?
Google's not really heard of it except for discussions of whether
it exists or not in the context of tracing the author of another
paper (by a B. Davis, rather than S. Davis).
Weird.
Whatever.
Phil
--
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Robin Chapman
> In article <20040114233049...@mb-m22.aol.com> hao...@aol.com
> (Dan Hoey) writes:
> > On Fri, 9 Jan 2004, Mark Griffith asked:
> > > Anyone know of any recent work on the odd perfect number
> > > question?
> >
> > Oddly enough, a paper claiming to prove the conjecture has
> > appeared, dated Thu, 8 Jan 2004, at
> > http://www.arxiv.org/abs/hep-th/0401052/
> > I haven't got the background to tell whether it's good or bad.
>
> If I correctly read what is written in that paper, the first error is
> around the top of the first page. I do not know whether 1.1 can be
> satisfied.
It's certainly one of the worst-written maths papers I've seen for a while.
He really is most allergic to actually explaining his notations. :-(
I nearly got to the end of section 1.
Abstract. He might mention that it's easy to prove that an odd
perfect number (indeed any odd number n with sigma(n) = 2 (mod 4))
has the form n = q^{4m+1} p_1^{2 r_1} ... p_s^{2 r_s}
where q, p_1, ..., p_s are distinct primes, and q = 1 (mod 4).
(He really should say that his 4k+1 is a *prime* distinct
from any of his q_i s).
Page 1. (1.1) is just a definition --- he's saying that
if []/[] is put into lowest terms, it's a_i/b_i.
(Here [] and [] are those gruesome fractions on either side
which I won't copy).
(1.2) is really an "if and only if". As it stands it's just
a complicated way of saying that N is *not* perfect.
(This is a bit loopy: his section heading talks of
a condition for a number being perfect so he writes down
one for a number being not perfect :-) )
I'm not sure why all those square roots are lying around...
why didn't he write this as 2(4k+1)(b_1 ...b_l)/() []^{l+1}.... =/= ... ?
(1.4) Now here his notation starts to get a bit gothic.
He has things like a_{13}. I reckon that doesn't mean the
thirteenth a_i but rather a_1 a_3. Also he has .Square .
It took me a while to realize that .Square means "times a square"
(presumably of a rational number). But realizing that, the formula
looks even more batty. Why include brackets like
(a_1 a_3 b_2/b_1 b_3 a_2) when he could have had
(a_1 a_2 a_3/b_1 b_2 b_3) without any difference in meaning?
After that he talks about these things, but in his notation he
shoves overlines onto his subscripts ... Why?
He then claims that some of these fractions aren't square
multiples of 2(4k+1) referring to his previous paper .....
In (1.5) he introduces some notation and talks about fractions being
"square-free".... why doesn't he keep things simple, and
represent these quantities as a square-free integer times
a square of a rational?
Next page... (1.7) lacks a parenthesis, but (1.6) through to (1.9)
are unreadable as they stand. E.g., (1.9) contains rho-hat_{2i,2}
which has never been defined. ((1.5) defines rho-hat_{3j+2}).
At this stage, I wonder whether there is any worth trying to
keep second-guessing this geezer as to what he means.
Anyway my guess as to the import of section 1 is that he reckons
with N of the form given in the abstract, sigma(N)/N can't
even be twice a square of a rational and he reckons he's proved
that in the sequel.
Robin Chapman
> hep-th = High Energy Physics, Theory,
>
> Hmmm... I thought General Math was the kooks' hangout?
>
> I'm glad I normally surround myself with mathematical matters, as this
> physics stuff, particularly the stuff Simon Davis is into, looks
> pretty heavy:
<snip>
> He seems in a very specialist field - "quantum cosmological wavefunction"
> yields only 19 hits in google, of which 18 pertain to Simon Davis' own
> papers.
>
> And what is the "research foundation of southern california"?
> Google's not really heard of it except for discussions of whether
> it exists or not in the context of tracing the author of another
> paper (by a B. Davis, rather than S. Davis).
Davis has 21 items on MathSciNet, all on stringy stuff apart from
his cited paper whose review I append.
MR1979400 (2004b:11007)
Davis, Simon(5-SYD-SM)
A rationality condition for the existence of odd perfect numbers. (English.
English summary)
Int. J. Math. Math. Sci. 2003, no. 20, 1261--1293.
11A25 (11B39 11D41 11D61)
Review in linked PDF Add citation to clipboard Document Delivery Service
Journal Original Article More links More links
References: 0 Reference Citations: 0 Review Citations: 0
If an odd perfect number exists, it must exceed $10^{300}$ \ref[R. P. Brent,
G. L. Cohen and H. J. J. te Riele, Math. Comp. 57 (1991), no. 196,
857--868; MR 92c:11004]. The author observes that an odd integer $N$ can be
perfect only if a certain product depending on its prime divisors has a
rational square root. This product contains factors which are "repunits",
that is, of the form $(q^n-1)/(q-1)$, for the primes $q$ appearing to an
even power in the canonical decomposition of $N$. By using this rationality
condition and properties of repunits the author studies the existence of
odd perfect numbers $N$. He obtains an upper bound for the density of such
$N$ in any fixed interval (above $10^{300}$) and proves the nonexistence of
$N$ in some special classes of integers. An important role in his
discussions is played by various Diophantine equations.
Reviewed by T. Metsänkylä
The maths department at Sydney University is certainly reputable.
The journal Int. J. Math. Math. Sci. is certainly legit
(they did once publish a paper by me, albeit a potboiler even
by my standards :-) )
A couple of the reviews of his stringy stuff include some barbed
comments:
"This article appeared in a physics journal and is written in the
corresponding style. ... The mathematical definitions of the above terms
are not given in the article under review; instead the author refers to his
previous work ..."
"... The author shows the "modular covariance" of the volume form
on Siegel's upper half-space. (This is of course a well-known
mathematical fact.) ..."
James Wanless
Well, one thing at least strikes me, which I don't think anyone else
has picked up on yet... [people seem to be more interested in his
affiliations and past papers - perhaps because they don't trust their
own _mathematical_ judgement :-)]
If you read his abstract (and concluding sentence) carefully it seems
to me he's only claiming anything (even leaving aside the merits of
his main argument) for odd numbers of the form 4k+1 [as opposed to
4k+/-1] (perhaps times a square). Therefore this completely lets slip
through the net numbers like 7, for example...
For the real proof of the non-existence of Odd Perfect Numbers see:
http://www.bearnol.pwp.blueyonder.co.uk/Math/perfect.html
J
Mateusz Kwasnicki
> For the real proof of the non-existence of Odd Perfect Numbers see:
> http://www.bearnol.pwp.blueyonder.co.uk/Math/perfect.html
> J
Well, I looked. Got some questions:
1. p is prime, is it not?
2. why p n == 0 [mod 2]?!?
Regards,
Mateusz
Phil Carmody
> hao...@aol.com (Dan Hoey) wrote in message news:<20040114233049...@mb-m22.aol.com>...
> > On Fri, 9 Jan 2004, Mark Griffith asked:
> > > Anyone know of any recent work on the odd perfect number
> > > question?
> >
> > Oddly enough, a paper claiming to prove the conjecture has
> > appeared, dated Thu, 8 Jan 2004, at
> > http://www.arxiv.org/abs/hep-th/0401052/
> > I haven't got the background to tell whether it's good or bad.
>
> Well, one thing at least strikes me, which I don't think anyone else
> has picked up on yet... [people seem to be more interested in his
> affiliations and past papers - perhaps because they don't trust their
> own _mathematical_ judgement :-)]
> If you read his abstract (and concluding sentence) carefully it seems
> to me he's only claiming anything (even leaving aside the merits of
> his main argument) for odd numbers of the form 4k+1 [as opposed to
> 4k+/-1] (perhaps times a square). Therefore this completely lets slip
> through the net numbers like 7, for example...
I fart more correct mathematical judgements than you.
Any odd perfect numbers must be of the form 4n+1, that's been known
for centuries.
> For the real proof of the non-existence of Odd Perfect Numbers see:
> http://www.bearnol.pwp.blueyonder.co.uk/Math/perfect.html
You're a freaking loon.
*_PLONK_*
Robin Chapman
> For the real proof of the non-existence of Odd Perfect Numbers see:
> http://www.bearnol.pwp.blueyonder.co.uk/Math/perfect.html
which says:
There exist no odd perfect numbers
Proof:
Suppose sigma(n)=2n [n perfect]
sigma(pn)=2n(p+1) [if hcf(n,p)=1]
p+1==0 [mod 2] [if p odd]
=> sigma(pn)==0 [mod 4]
Let p->99999... [Euclid]
sigma(pn)/pn->2
=> pn==0 [mod 2]
=> n==0 [mod 2] [since p odd]
Copyright 1997 James Wanless
oops! I copied it :-)
So what can we tell from this?
That we can conclude that as p -> 99999... [Euclid]
(dunno what "99999... [Euclid]" is but never mind) and
sigma(pn)/pn -> 2 then pn is even. Hmmm. I ask Mr Wanless this?
Does (6p+1)/3p -> 2 as p -> 99999... [Euclid], and does
it follow that 3p is even (even if it's odd?)?
mark griffith
I published my own naive geometrical argument that there are no odd perfect numbers on Jan 14th here http://mathquest.com/discuss/sci.math/m/569936/570941
and my non-standard notation irritated a reader or two - apologies.
My outline argument is that since Euler showed an odd perfect number must be a multiple of a square, then if we consider how summing the factors must "fill in" an array of squares equalling the odd prime, a contradiction emerges when we look at the square holding some or all of the subfactors of the factor which is a square [which I helpfully name 'sq'].
If anyone friendly has the patience to wade through three pages of that, I'd be grateful for an opinion, and will gladly answer any questions.
Thanks, Mark G.
Artyom_ch
Extremely hard to get your ideas. However, I tried my best.
Comments follow.
>Sorry about delay.
>
>.
>
>Since we know that the first odd perfect number [OP]
>must be a product of a square [sq] and an odd power of
>a single prime [pr] we start with the case where pr is
>raised to the power of 1 and sq.pr = OP.
>
>
>(1) Visualise OP as an array of pr-many squares, each
>containing sq-many tiles, and as OP is perfect we
>assign all factors as follows:
>
>i] one square represents the factor sq itself;
>ii] pr-2 squares contain factors of sq multiplied by
>pr;
>iii] one square contains factors of sq + an overflow
>from [ii] which we call x.
>
>We know that there is an overflow from the factors of
>sq multiplied by pr, since [a] the factors of sq
>cannot sum to more than sq [else the factors of OP
>will sum to more than sq.pr], nor can they sum to
>exactly sq since by hypothesis OP, not sq, is the
>first odd perfect and also the factors of sq must have
>an even sum [odd-many proper factors + 1].
>
>Therefore the factors of sq sum to sq-x where x is
>odd. Further, x must be 1 or else divide into sq,
>since [sq-x]pr = [pr-2]sq + x and so x.pr = sq + sq-x.
>
>
>(2) But by the following argument the factors of an
>odd square cannot sum to sq-x where x divides into sq.
>
>Either x=1 or x>1. Suppose x=1.
To complex. For x=1 proof takes 2 lines :)
>
>We know that factors which are paired with
>whole-number multiples of sqrt will occupy the same
>number of tiles in one column as the factors which are
>whole-number multiples of sqrt will occupy whole
>columns elsewhere.
>
>So if x=1 then sq-1 takes up (sqrt-1)(sqrt + 1) tiles.
>We can either divide the factors into two groups, one
>of which (larger) sums to (sqrt-1).sqrt and the other
>(smaller) to (sqrt-1), or we cannot.
>
>If not, then one larger factor either overlaps or
>underlaps the divide between the (sqrt-1).sqrt area
>and the (sqrt-1) area. In either case there is a
>contradiction, since the larger-factor sum is either
>too large to be sqrt times the smaller-factor sum
>(overlapping), or too small to be sqrt times the
>smaller-factor sum (underlapping).
>
>So we must have a set of larger factors which sum to
>an exact multiple of sqrt. But dividing the sum of
>larger factors through by sqrt should then give a
>whole number, sqrt-1. We note that with cases like sq =
>abb.abb we get larger factors like bbb and bbbb and
>with cases like sq = abc.abc larger factors like aabb,
>aacc, bbcc which cannot sum to an integer total when
>divided by sqrt. By induction on numbers being
>multiplied up into products with more factors, we note
>that as xy can only multiply up to xyy or to xyz with
>the addition of one factor then we will always have
>incommensurable sums when dividing by sqrt.
>
>Therefore the larger factors which are not already whole
>multiples of sqrt will not sum to a whole multiple of
>sqrt+1. Although sq-1 divides without remainder by
>sqrt+1, we cannot divide the sum of factors into two
>blocks measuring sqrt-1 and [sqrt-1].sqrt in size.
>
>Therefore no odd squares have factors summing to
>exactly sq-1.
>
>
>(3) This leaves x>1 where x divides into sq.
>
>Dividing sq-1 by sqrt+1 gives sqrt-1, so we know that,
>since sqrt is odd, that there is an even multiple of
>sqrt+1 in sq, with a final remainder of 1.
>
>To test whether the factors of sq could sum to sq-x
>where x divides into sq, we divide sqrt+1 both into
>sq-x and into x and examine the remainders, r1 and r2.
>Let [sq-x]/[sqrt+1] = p + r1, and x/[sqrt+1] = q + r2.
I suppose you mean sq-x = p(sqrt+1)+r1, x=q(sqrt+1)+r2.
(from your variant of equations r1 and r2 are not integers)
>
>Since there is still the final remainder of 1 to
>complete sq, if overflow x can divide into sq and
>complete sq, then r1 + r2 must equal sqrt + 1 + 1.
>
>As sqrt+1 cannot divide into sq, so not into sq-x or x
>either, we know both r1 and r2 are non-zero.
>
>But since sq-x is even, x is odd, and p + q odd [the
>total number of sqrt+1 in sq-1 is even = sqrt-1 = p +
>q + 1] we either have an odd p leaving an odd
>remainder in sq-x and an even q leaving an odd
>remainder in x, or an even p leaving an even remainder
>in sq-x and an odd q leaving an even remainder in x.
>Either way, r1 + r2 is even, but sqrt + 2 is odd.
Here is my main question.
sq-x = p(sqrt+1)+r1, x=q(sqrt+1)+r2 (previous comment).
sq-x -- even, sqrt+1 -- even. Then r1 is even with any p( from 1st
equation).
x -- odd, sqrt+1 - even. Then r2 is odd with any q.
Then r1+r2 always odd. But you wrote opposite statement.
>
>Therefore the factors of an odd square sq cannot sum
>to sq-x if x>1 and divides into sq.
>
>Since this obstacle holds equally for higher powers of pr,
>we see there are no odd perfect numbers.
>
>Mark Griffith
>
>14th Jan 04
>
>.
>