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[math-fun] claim of proof for No Odd Perfect Numbers
by Richard Schroeppel 21 Jan '04

21 Jan '04

Paper (*cross-listing*): hep-th/0401052 From: Simon Davis Dr <davis(a)math.uni-potsdam.de> Date: Thu, 8 Jan 2004 23:56:42 GMT (9kb) Title: A Proof of the Odd Perfect Number Conjecture Authors: Simon Davis Comments: TeX, 13 pages Report-no: RFSC 04-01 Subj-class: High Energy Physics - Theory; Number Theory The finiteness of the number of solutions to the rationality condition for the existence of odd perfect numbers is deduced from the prime decomposition of the product of repunits defined by the sum of the divisors, $\sigma(N)$. All of the solutions to this rationality condition satisfy the inequality ${{\sigma(N)}\over N}\ne 2$. This technique is then used to demonstrate that there are no odd perfect numbers $N=(4k+1)^{4m+1} \prod_{i=1}^\ell q_i^{2\alpha_i}$. http://arXiv.org/abs/hep-th/0401052 13 pages ---------------------- I looked at this, but haven't puzzled through it yet. A crank? Putting the paper in hep-th is curious, and the author affiliation is unusual. The surface markers look mostly reasonable: real references, real notation. The logic is abbreviated, and it's hard to tell if it's merely explanatory. Page 1 leaps immediately into complicated algebra, eschewing background. There are well known required shapes for hypothetical odd perfect numbers. (These arise from equating 2N with the formula for the sum of divisors of N, and analyzing the even/odd properties of base P repunits. "Elementary" number theory.) One requirement is that all but one prime divisor have even exponent, making a square of a prime or prime power. One part of Davis's argument is that the divisor sum term for these primes (a factor of P^2 leads to a divisor-sum factor of P^2 + P + 1, a base P repunit 111) leads to new primes > P, which must then appear as new divisors of the hypothetical perfect number. For example, 7^2 -> 7^2 + 7 + 1 = 57 = 3 * 19 giving the new prime 19. Continuing the argument, 19^2 -> 381 = 3 * 127, and so on. If this always led to new larger primes, it would prove NoOPN. But it doesn't always give new primes (67^2 -> 4557 = 3 * 7 * 7 * 31 and 79^2 -> 6321 = 3 * 7 * 7 * 43), and I think his argument is concerned with handling the exceptional cases. Rich rcs(a)cs.arizona.edu

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[math-fun] Fermat's Last Theorem
by Jon Perry 21 Jan '04

21 Jan '04

Excuse my ignorance please, but wrt to A. Wiles proof of FLT, is the proof valid for rationals and/or with rational powers? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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Re: [math-fun] When liberals do science...
by Henry Baker 20 Jan '04

20 Jan '04

I'm not quite sure that Chernobyl is a liberals v. conservatives thing. I don't recall anyone thinking that vastly increased radiation is a good and/or natural thing (although zero radiation may be a bad thing, because the body's repair mechanisms never get triggered, and evolution may be slowed down). Even pro-nuke people fully expected a Chernobyl-type incident, because the Chernobyl-type reactors fail in very bad ways. So the bad guys here were _communists_, not conservatives. Is someone here suggesting that there are nasty Chernobyl-type causes of global warming? Perhaps something going on in Area 51 in Nevada? The problem with global warming is that the Earth is a very complex system involving many negative-feedback loops (the good kind, if you want to keep things more-or-less constant). The Gaia hypothesis says that life itself has modified the Earth's environment to better favor life and buffer it from severe attacks from space in the form of radiation, changes in the Sun's output, asteroids, etc. If we took the arguments of the tree-huggers to their ultimate conclusion, we would get rid of all oceans and all trees and all oxygen to put the Earth back to its "natural" state -- something more akin to Mars/Venus/the Moon. Perhaps Mars was the Sierra Club's/Earth First's original success? It is now becoming clear that "preserving forests from fire" is worse than self-defeating, as well as entirely unnatural. Those of us who live in the West which has been devastated by fires know first-hand how the delusions of these East Coast tree-huggers have been killing off the forests in ways far worse than any logging could possibly have done. Instead of the mild fire every 10-20 years, we now have nuclear winter-type forest fires caused by 100 years of "suppression". It now appears that a proper way to deal with forest fires might be to lay out a geometric grid pattern of some type, color it with 6-8 colors, and then burn the part of the forest of that color each 6-8 years _on purpose_. If properly laid out, the burns of previous years would be natural firebreaks, and by burning on a shorter schedule, it wouldn't be possible to build up enough fuel to get the temperatures so hot as to kill all the larger trees -- e.g., the sequoias. Such smaller, cooler burns also recover much faster -- usually within 1 year. If you want to reduce the amount of carbon dioxide produced by such burns, then perhaps the best thing to do would be to carefully log them first, so there is less to burn. The one thing you won't be able to do using any technology available for the next 100 years is to stop forest fires entirely. We've proved that conclusively -- they still occur, they still put just as much CO2 into the atmosphere, but they are completely uncontrolled. *** This is not a human activity. These fires have been going on for hundreds of millions of years. Get over it. *** At 10:06 AM 1/19/2004, John Conway wrote: > I well remember being in England about 20 years ago when the whole of >Western Europe was suddenly very much concerned at the rapid rise in >radioactivity that we later learned was the result of an accident at >Chernobyl in Russia (but now Ukraine once again).

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Re: [math-fun] Fun from numbers of groups of given orders.
by Christian G. Bower 20 Jan '04

20 Jan '04

John Conway <conway(a)Math.Princeton.EDU> wrote: ... > This raises the interesting question of whether every positive integer > is the number of groups of some order? O'Beirne told me that Keith > Devlin has worked on this, but I didn't get the impression he'd got > very far. I remember corresponding with Keith Dennis on that very question a few years ago. I contacted him after he submitted the sequence: > Most of my thinking has been on the universality question - > "is every positive integer the number of groups of some order?". > As I said, I'll report on that later. It gives rise to some more > interesting sequences by the way, including 7,11,19,29,31,47,49,... . A053403 in the EIS

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Re: [math-fun] Fun from numbers of groups of given orders.
by MCKAY＠vax2.concordia.ca 20 Jan '04

20 Jan '04

John - is this your dotage? Does it have much interest? I have wanting to study here, a Bangladeshi student who published the number n for which there exists just one group of order n. Not very exciting ? Recovered from op and NZ? How as NZ? Best, John From: IN%"math-fun(a)mailman.xmission.com" "math-fun" 20-JAN-2004 14:39:47.80 To: IN%"math-fun(a)mailman.xmission.com" "math-fun" CC: Subj: [math-fun] Fun from numbers of groups of given orders. Return-path: <math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com> Received: from clyde.concordia.ca (clyde.Concordia.CA [132.205.1.1]) by vax2.concordia.ca (PMDF V6.2 #30759) with ESMTP id <01L5MIWZIG14002VIR(a)vax2.concordia.ca> for mckay(a)vax2.concordia.ca (ORCPT mckay(a)vax2.concordia.ca) Tue, 20 Jan 2004 14:39:47 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by clyde.concordia.ca (8.12.10/8.12.10) with ESMTP id i0KJdDkE332091 for <mckay(a)vax2.concordia.ca>; Tue, 20 Jan 2004 14:39:31 -0500 (EST) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1Aj1YQ-0004TR-0D for <mckay(a)vax2.concordia.ca>; Tue, 20 Jan 2004 12:28:50 -0700 Received: from mail.math.princeton.edu ([128.112.18.14]) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1Aj1YH-0004TL-00 for <math-fun(a)mailman.xmission.com>; Tue, 20 Jan 2004 12:28:41 -0700 Received: from fine318a.math.Princeton.EDU (fine318a.math.princeton.edu [128.112.17.139]) i0KJSdDq015954 (version=TLSv1/SSLv3 cipher=EDH-RSA-DES-CBC3-SHA bits=168 verify=NOT) for <math-fun(a)mailman.xmission.com>; Tue, 20 Jan 2004 14:28:40 -0500 Received: from fine318a.math.Princeton.EDU (localhost.localdomain [127.0.0.1]) i0KJSdrA014191 for <math-fun(a)mailman.xmission.com>; Tue, 20 Jan 2004 14:28:39 -0500 Received: from localhost (conway@localhost) i0KJScJN014187 for <math-fun(a)mailman.xmission.com>; Tue, 20 Jan 2004 14:28:38 -0500 Date: Tue, 20 Jan 2004 14:28:38 -0500 (EST) From: John Conway <conway(a)Math.Princeton.EDU> Subject: [math-fun] Fun from numbers of groups of given orders. In-reply-to: <200401141942.i0EJgr925416(a)baskerville.CS.Arizona.EDU> Sender: math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com To: math-fun <math-fun(a)mailman.xmission.com> Errors-to: math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com Reply-to: math-fun <math-fun(a)mailman.xmission.com> Message-id: <Pine.LNX.4.44.0401201344070.14164-100000(a)fine318a.math.Princeton.EDU> MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII Precedence: list X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.1 X-Scanned-By: MIMEDefang 2.35 List-Post: <mailto:math-fun@mailman.xmission.com> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Id: math-fun <math-fun.mailman.xmission.com> Original-recipient: rfc822;mckay(a)vax2.concordia.ca While in New Zealand I met Eamonn O'Brien, who's one author of a paper that, among other things, tabulates the number of groups of each order up to 2000. This has led me to find quite q few interesting sequences, and given quite a lot of fun. First, some comments on the numbers. For n = 1 to 32, the number of groups of order n is 1,1,1,2,1,2,1,5,2,2,1,5,1,2,1,14,1,5,1,5,2,2,1,15,2,2,5,4,1,4,1,51. Certain numbers (such as 1,2,5,15) are much more common than others. The numbers of groups of orders 1,2,4,8,16,32,64,128,256,512,1024 are 1,1,2,5,14,51,267,2328,56092,10494213,49487365422. The number of groups of order at most 2000 but not 1024 is only 423164062, so the ones of order 1024 outnumber the others 10-to-1! Remarkably, the number of groups of order p^n (p prime) is known quite closely - it has the form p^{ (2n^3/27) + O(n^{5/2}) }, the 5/2 here improving on an earlier bound of 8/3. The smallest N for which the number of groups of order N is 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 is 1 4 75 28 8 42 375 510 308 90 140 88 56 16 24 100 675 156 1029 820 1875 For n = 22, we must have N > 2000. This raises the interesting question of whether every positive integer is the number of groups of some order? O'Beirne told me that Keith Devlin has worked on this, but I didn't get the impression he'd got very far. Lenstra and I pushed it quite a long way, though with some mistakes that I've since been rectifying. I'll report on this later. I propose to use GROUPS(N) for the number of groups of order N, but will abbreviate it to G(N) in this letter. Let's call N group-deficient or group-abundant according as G(N) < N or G(N) > N, and group-perfect if G(N) = N. Then it seems almost certain that 1 is the only group-perfect number, and that almost all numbers are group-deficient. The best we've proved in this direction, though, is that all squarefree numbers (and so most numbers) are group-deficient. Here's the elegant proof that was found by Lenstra and simplified a bit by me: Any group of squarefree order N has a presentation of the form 1 = a^d = b^(N/d), ab = ba^k, where d is some divisor of n, and k is prime to d. So the number of such groups is at most the sum of phi(d) over d dividing N, which is well-known to be N. However, if N > 1 this counts the cyclic group twice (for d = 1 and d = N), and so then the number of groups of order N is strictly less than N. One of the conjectures above can be put in form A(N)/N -> 0, where A(N) is the number of group-abundant numbers up to N. However, the convergence is slow. For example, if N = 49487365422 and n = 1024, then every multiple of n below N is group-abundant, and so A(N)/N exceeds 1/n. We can replace n by any 2^m > 16 and N by GROUPS(n), which is approximately 2^{(2n^3)/27}, to see that this argument proves that we have 1 A(N)/N > ------------------------- roughly, 3 times cuberoot(logN /2) for an infinity of N, the logarithm being taken to base 2. Here's how the sequence of group-abundant numbers starts: 1,32,48,64,96,128,144,160,192,256,288,320,384,432,448,480,512,576,640,648, 672,704,720,768,800,832,864,896,960,1024,1088,1152,1216,1248,1280,1296, 1344,1408,1440,1458,1536,1600,1664,1728,1792,1920,1944 Lenstra's theorem tells us that none of these can be squarefree - in fact so far they've always been divisible by some fourth power, which suggests that perhaps that theorem can be improved. Most of my thinking has been on the universality question - "is every positive integer the number of groups of some order?". As I said, I'll report on that later. It gives rise to some more interesting sequences by the way, including 7,11,19,29,31,47,49,... . REgards to all - JHC _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] When marketroids do science...
by Mike Stay 20 Jan '04

20 Jan '04

http://www.minimed.com/patientfam/pf_ipt_paradigm_insulin_pump.shtml RF Technology Provides More Flexibility RF technology uses sound waves to "beam" information instead of light waves like infrared (IR) technology. To understand why RF is more adaptable, compare your TV remote control to a cell phone. Your remote control uses infrared (IR) technology, so you must point it directly at your TV to beam the light waves. Cell phones, however, use sound waves that travel more freely, even penetrating solid objects. As a result, you eliminate the inconvenience. -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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Re: [math-fun] Image understanding
by Tom Knight 20 Jan '04

20 Jan '04

I'd recommend the book "The Fourier Transform and its Applications" by Ron Bracewell. It has a good section on 2-D transforms, and is, in general, an excellent text.

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[math-fun] Image understanding
by The Purple Streak, Hilarie Orman 20 Jan '04

20 Jan '04

Is there something like "elementary introduction to FFT's for digital photography" ? I'm trying to develop (!) some heuristics for automatically processing my digital photos, and I'm doing FFT's on them to try get a sense the orientation. Few of my photos are of fences or zebras, so I need some experience in understanding how to interpet FFTs of general scenes. Hilarie Wears Full Bark Jacket for tree hugging Help Stomp Out Box Elder Beetles

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Re: [math-fun] Tenuous Associations
by Mike Stay 19 Jan '04

19 Jan '04

Numerology. For a wonderful treatment of the subject, along with Templars, Rosicrucians, and the like, read _Foucault's Pendulum_ by Umberto Eco. ----- Original Message Follows ----- > I have in front of me an article "The Mystery of Time" > from "The Mongol Messenger", a Mongolian newspaper out of > Ulaan Bataar, dated January 2002. This article was sent > to me by a friend in the country, who thought I might find > it of mathematical interest. The content of the article > is quite entertaining, though of little mathematical > merit. The author is apparently from the UK. > > I would consider the article a crackpot piece, though it > appears to be well written and of a less confrontational > tenor than many such articles I have read. It contains > some interesting equations(sic). It is difficult to > summarize the article, which seems to be a hodgepodge of > mystical and mathematical associations revolving around > the concept of time, but with no overriding premise or > point. I have a mosaic scan of the article available upon > request, about 6MB. > > The author seems to be fixated on certain historical dates > , 1543 (Copernicus's heliocentric theory), 1618 (Kepler's > laws), 1687 (Newton's Principia), 2001 (unified physics?), > and 2060 (Newton's Last Judgment date?). The Hindu cycle > (4320000000 years?) also figures prominent. All sorts of > wondrous mathematical associations are used to justify or > explain the significance of these dates. > > The types of things done in this article are common in > crackpot literature, particularly the misuse of > associations. Since mathematics is so rich in > associations, and numbers are so pervasive in our affairs, > its is only natural that unjustifiable mathematical > associations between unrelated numbers would suggest > themselves. And if you push it, you can find "significant" > relationships between all sorts of unrelated numbers. > > I was wondering if there is a name for the practice of > drawing tenuous assocations between unassociated entities > from a rich association domain, and specifically, the > practice of drawing mathematical associations between > mathematically unrelated numbers? > > For instance, let us suppose I see special historical > significance in the person of George Washington (which I > do). I notice that his birth year, 1732 is reminiscent of > sqrt(3) = 1.732+, and from that propose some sort of > mystic significance. So let us drag down this mystic > association. > > First, we assume that George Washington's birth date is a > significant determinant or indicator of his personal > signficance, not an uncommon practice, certainly a major > underpinning of astrology and other mystical arts. Then > we assign his birth year the number 1732. This is a > customary, but ultimately arbitrary number. It indicates > that the Earth has orbited the sun somewhere between 1732 > and 1733 times between the unknown and probably incorrect > time of the birth of Jesus Christ and the time of birth of > George Washington. 1732 has a visual similarity to the > number 1.732, an approximation to the square root of 3. > The visual similarity rests on the peculiarities of our > decimal notation, which itself rests on the number 10, > presumably the number of fingers on our two hands. The > actual mathematical association is 1732 = floor(1000 * > sqrt(3)). > > I could take this exploration further, but suffice it to > say that this obvious mystic significance I have assigned > to George Washington is based on a not-so-simple > mathematical relationship between several unrelated > quantities, including the time of George Washington's > birth, the time of Jesus Christ's birth, errors in the > calculation of the latter date, the period of revolution > of the Earth about the Sun, the number of fingers on the > hands of human beings, and the arbitrary integers 3 and > 1000. To which of these quantities does the relation 1732 > = floor(1000 * sqrt(3)) lend significance? And certainly > there are many other birth dates and events occuring in > the year 1732 as we reckon it, which were not nearly so > historically significant as the birth of George Washington > , which leads us to question the force of the association. > > Is there a name for the inappropriate use of tenuous > associations such as the one described above? If not > there should be. Because numbers are rife with > mathematical associations, and are so pervasive in our > everyday experience, they lend themselves to this sort of > abuse, and there should be a specific term for this type > of abuse of numerical associations. > > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] Mixed Doubles
by R. Hess 19 Jan '04

19 Jan '04

A happy new year problem for math funsters. Tennis -- Mixed Doubles Round Robin 5 courts in play 10 men(M), 10 women(W) (say the players are numbered 1-20, the men being odd and the women even) All matches are M/W vs M/W No person partners any person more than once. No person opposes any person (M or W) more than once. I have a solution for 5 rounds (25 matches) obeying the above requirements. Is it possible to do more rounds? What is the general limit for 2m men and 2m women playing mixed doubles? Happy puzzling, Dick Hess

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