May 2004 - mathfun@mailman.xmission.com

Re: [math-fun] cicada periods
by MCKAY＠vax2.concordia.ca 27 May '04

27 May '04

There are plenty of places where the cicadas do not have these long periodicities. Jm From: IN%"math-fun(a)mailman.xmission.com" "math-fun" 26-MAY-2004 11:14:30.34 To: IN%"math-fun(a)mailman.xmission.com" "math-fun" CC: Subj: RE: [math-fun] cicada periods Return-path: <math-fun-bounces+mckay=vax2.concordia.ca(a)mailman.xmission.com> Received: from bonnie.concordia.ca (bonnie.Concordia.CA [132.205.7.81]) by vax2.concordia.ca (PMDF V6.2 #30759) with ESMTP id <01LAJQRBP2IO0089BI(a)vax2.concordia.ca> for mckay(a)vax2.concordia.ca (ORCPT mckay(a)vax2.concordia.ca) Wed, 26 May 2004 11:14:29 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by bonnie.concordia.ca (8.12.10/8.12.10) with ESMTP id i4QFERA0023845 for <mckay(a)vax2.concordia.ca>; Wed, 26 May 2004 11:14:28 -0400 Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BT06t-0005gi-03 for <mckay(a)vax2.concordia.ca>; Wed, 26 May 2004 09:14:27 -0600 Received: from mgr8.xmission.com ([198.60.22.208] ident=mail) by mailman.xmission.com with esmtp (Exim 3.36 #1 (Debian)) id 1BT06m-0005gb-00 for <math-fun(a)mailman.xmission.com>; Wed, 26 May 2004 09:14:20 -0600 Received: from [24.75.96.125] (helo=smtp.firstva.com) by mgr8.xmission.com with esmtp (Exim 4.31) id 1BT06j-0004XQ-T0 for math-fun(a)mailman.xmission.com; Wed, 26 May 2004 09:14:20 -0600 Received: from [4.240.87.156] (helo=nexet.net) by smtp.firstva.com with asmtp (Exim 4.24) id 1BSzuI-0005kO-4o for math-fun(a)mailman.xmission.com; Wed, 26 May 2004 11:01:26 -0400 Date: Wed, 26 May 2004 09:13:28 -0500 From: scott beaver <sbeaver(a)nexet.net> Subject: Re: [math-fun] cicada periods In-reply-to: <005001c442e5$7dc84f40$a52efea9@TASSO> 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: <40B4A608.8080400(a)nexet.net> MIME-version: 1.0 Content-type: text/plain; format=flowed; charset=us-ascii Content-transfer-encoding: 7bit X-Accept-Language: en-us, en Precedence: list User-Agent: Mozilla/5.0 (Windows; U; Win98; en-US; rv:1.5) Gecko/20031007 X-Spam-Checker-Version: SpamAssassin 2.63 (2004-01-11) on mgr8.xmission.com X-Spam-Level: X-Spam-Status: No, hits=-0.9 required=8.0 tests=BAYES_30 autolearn=no version=2.63 X-SA-Exim-Connect-IP: 24.75.96.125 X-SA-Exim-Mail-From: sbeaver(a)nexet.net X-SA-Exim-Version: 4.0 (built Sat, 24 Apr 2004 12:31:30 +0200) X-SA-Exim-Scanned: Yes (on mgr8.xmission.com) X-BeenThere: math-fun(a)mailman.xmission.com X-Mailman-Version: 2.1.4 X-Scanned-By: MIMEDefang 2.35 References: <005001c442e5$7dc84f40$a52efea9@TASSO> 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 Isn't interbreeding a survival strategy? Scott Thane Plambeck wrote: >>From Nature Magazine, 20 May 2004, "Big Buzz as cicadas arrive after 17-year gap" > * * * > Researchers say that in some places there will be more than 370 noisy, colorful insects > per square meter. The scene will be "like a science fiction movie," enthuses May Berenbaum, > who is an entomologist at the University of Illinois at Urbana-Champagne... > > [...] > > ...But perhaps the most interesting question about cicadas is why they spend 17 years > underground, a cycle that has been recorded since the mid-nineteenth century. > > Biologists think it is no coincidence that cicadas have evolved to reappear with periods > that are large prime numbers. If the life cycle of a breed were 12 years, they speculate, > it would interbreed frequently with others that had life cycles of 2,3,4, or 6 years. "The > prime number prevents mating between two broods and hybridization,'' says Christine > Simon, an evolutionary biologist at the university of Connecticut. There are 13- and 17- > year broods, and these can only meet up every 221 years, giving them ample time > to forge their unmistakable identities > * * * * > "So many cicadas, so few recipes" (I read that somewhere else, forgot where) > > Thane Plambeck > 650 321 4884 office > 650 323 4928 fax > http://www.plambeck.org > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > _______________________________________________ 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] cicada periods
by Thane Plambeck 26 May '04

26 May '04

>From Nature Magazine, 20 May 2004, "Big Buzz as cicadas arrive after 17-year gap" * * * Researchers say that in some places there will be more than 370 noisy, colorful insects per square meter. The scene will be "like a science fiction movie," enthuses May Berenbaum, who is an entomologist at the University of Illinois at Urbana-Champagne... [...] ...But perhaps the most interesting question about cicadas is why they spend 17 years underground, a cycle that has been recorded since the mid-nineteenth century. Biologists think it is no coincidence that cicadas have evolved to reappear with periods that are large prime numbers. If the life cycle of a breed were 12 years, they speculate, it would interbreed frequently with others that had life cycles of 2,3,4, or 6 years. "The prime number prevents mating between two broods and hybridization,'' says Christine Simon, an evolutionary biologist at the university of Connecticut. There are 13- and 17- year broods, and these can only meet up every 221 years, giving them ample time to forge their unmistakable identities * * * * "So many cicadas, so few recipes" (I read that somewhere else, forgot where) Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] Article in NY Times on math
by Dan Asimov 26 May '04

26 May '04

I'm curious to learn people's opinions of the essay in today's NY Times science section at <http://www.nytimes.com/2004/05/25/science/25math.html>, "When Even Mathematicians Don't Understand the Math". --Dan

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RE: Re[2]: [math-fun] Re: Polynomial primogeniture
by Dan Asimov 26 May '04

26 May '04

Richard Guy writes: << ... what is significant is not the actual density over the first so many values, which clearly has to tend to zero in all cases, but the {\bf asymptotic} density, which, if we believe Hardy \& Littlewood (see {\bf A1}), is always $c\sqrt n/\ln n$, and the best that can be done \hGidx{asymptotic density} is to make the value of $c$ as large as possible. ... >> Yes, that's exactly what I'm interested in -- the asymptotic behavior. Richard: Is the Hardy-Wright asymptotic density of c sqrt(n) ^ (ln (n)) (please confirm that my parentheses are properly placed!) specifically for *quadratic* polymonials, or for *all* polynomials? Thanks, Dan

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[math-fun] Re: latest UPiNT
by Richard Guy 26 May '04

26 May '04

Here's an excerpt from the 1981-08-13 Preface. The idea for the book goes back some twenty years, when I was impressed by the circulation of lists of problems by the late Leo Moser and co-author Hallard Croft, and by the articles of Erd\H{o}s. Croft agreed to let me help him amplify his collection into a book, and Erd\H{o}s has repeatedly encouraged and prodded us. After some time, the Number Theory chapter swelled into a volume of its own, part of a series which will contain a volume on Geometry, Convexity and Analysis, written by Hallard T.~Croft, and one on Combinatorics, Graphs and Games by the present writer. The Geometry volume (mainly due to Ken Falconer, a student of Croft) appeared in 1991 and UPINT2 in 1994. UPINT3 (still a single volume, but now with about 450 pages) should be out already, but Springer seem to be having trouble making Andy Guy's old program work. I've been able to make it work for at least six months, during which time I've continued to update it. I've had to call a halt for the last month, though. I could send you a hard copy of the cover, which Springer have already issued! R. On Tue, 25 May 2004, Harvey Dubner wrote: > Richard, > > Has a new edition of UPiNT been published recently? Is something planned > for the near future? > Is there more than a Volume 1? > > Could you give me a short summary of its hisory? Someone asked me and I was > at a loss for facts. > > Harvey Dubner

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RE: [math-fun] Re: Polynomial primogeniture
by Dan Asimov 26 May '04

26 May '04

I (well, OK, a C program) factored Q(N) = N^2+N+1 for 10000 <= N < 11000, and found there to be 128 primes among these 1000 numbers. Note: x/ln(x) at Q(10^4) is ~ 5429194, whereas x/ln(x) at Q(10^4) + 1000 ~ 5429245. So the Prime Number theorem estimates about 51 primes for that run of 1000 consecutive numbers. This should be an upper bound, with high probability, for the # of primes among 1000 integers K selected randomly with Q(10^4) <= K < Q(10^4+10^3). But the actual count is 128, about 5/2 that number. Can the absence of prime factors of the form 6k-1 adequately explain such a high number of primes? --Dan A. -------------------------------------------------------------------- Don Reble djr(a)nk.ca writes: I wrote: > > ... factorization of Q(N) = N^2+N+1 > > ... > > there appears to be an unusually high frequency > > of Q(N)'s being prime. And also when Q(N) > > is composite, it appears as though there tend > > to be unusually few factors (and unusually large ones). > > > > Is there a theory that would confirm these suspicions? > > I might know a little bit about that... > > Q is the third cyclotomic polynomial, a factor of N^3-1, and Q(N) > can't be divisible by a 6k-1 prime. Since there are fewer prime > factors to choose from, you get fewer factors, and more primes. > See also http://www.asahi-net.or.jp/~KC2H-MSM/cn

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[math-fun] Mersenne Numbers - doh!
by Guy Haworth 26 May '04

26 May '04

Thanks to Don Reble for the composite M(p), and to Richard Schroeppel for the gentle reminder of the (2p+1)|M(p) theorem. It's obviously been too long since I looked at my own notes of the '80s. Curious that M(M(61)) hasn't succumbed to factor-searches yet. There must be a good chance that it's prime. http://www.ltkz.demon.co.uk/ar2/mm61.htm may or may not still be active. "Last updated 2001" - mmm. Guy

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Re: [math-fun] Article in NY Times on math
by Dan Asimov 25 May '04

25 May '04

Gene, I'm not sure what "genuine mysticism" means, but I suspect that whatever it means, I'd almost certainly agree with you that it's nonsense. Just between you and me, though: At the age of 21, when I was (much like today) 100% non-religious and 100% skeptical of anything lacking a firm grounding in evidence, I had an experience that was so remarkable that if my closest friend had had that experience instead of me, and had sworn to me that he'd had it, I'm sure I would not have believed him. It was such a totally different experience from anything else I'd ever heard of, no less experienced, that I never mentioned it to anyone else for a full year. About then, I picked up a friend's copy of William James's book "The Varieties of Religious Experience" and happened to read the chapter called "The Mystical Experience". So accurately was my experience described there, that when I read that chapter, I knew that other people had gone through essentially the same thing. I still have no idea (outside of a few wild, totally untestable possibilities) what to make of that experience. And I certainly did not become religious as a result of it. But I can assure you that the experience that James describes in that chapter is not fiction. (I hope you won't mind keeping this just between us.) --Dan ---------------------------------------------------------------------------- ------------------------------------ > The target audience for the books discussed in the this article > consists of people who are looking for a mystical experience, but who > are astute enough to appreciate that genuine mysticism is nonsense.

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[math-fun] WireWorld Multiplication
by Ed Pegg Jr 25 May '04

25 May '04

My latest Math Games column contains some interesting methods for multiplication. Nyles Heise showed how EF4E75E7×EFA03229 could be calculated in a 22×93 rectangle with WireWorld logic. I had no idea that multiplication could be done so efficiently. The full article is here: http://www.maa.org/editorial/mathgames/mathgames_05_24_04.html The column includes a very nice article by Brian Silverman, The Virtual Computer. At http://www.mathpuzzle.com/, I show a new crossword discovery ... an 8×9 grid of french words reading across and down, found by Jean-Charles Meyrignac: D E C R O C H E S E C O E U R A N T R O N F L A N T E A T T R I S T E R P A R A P H E R A E M A C I E R A S R E I T E R A I S A S S E N A S S E I recently heard from Don Albers about the Martin Gardner Mathematical Games collection. They had to secure an enormous number of permissions, which delayed things, but he fully expects maa.org will be able to release the collection in late summer. For some reason, the following comment didn't go through, in response to "An intriguing continued fraction cubic", a while back: I like fourth root of 9.1. The first 52 terms of the continued fraction are: 1, 1, 2, 1, 4, 75656, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 136181, 1, 2, 4, 1, 64, 1, 1, 3602, 4, 1, 12, 7, 8, 1, 2, 4267, 2, 9, 1, 22, 1, 1, 1, 1, 1, 1, 1, 4841, 35, 1, 5, 5, 1, 262344 ... --Ed Pegg Jr

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[math-fun] Polynomial primogeniture
by Dan Asimov 24 May '04

24 May '04

Just for fun, I decided to look at the factorization of Q(N) = N^2+N+1 as N gets large. So far I've just looked at 1 <= N <= 1000, but there appears to be an unusually high frequency of Q(N)'s being prime. And also when Q(N) is composite, it appears as though there tend to be unusually few factors (and unusually large ones). Is there a theory that would confirm these suspicions? More generally, are there integer polynomials P(N) -- taking positive values on positive integers -- such that the frequency of prime values as N -> oo is unusually large (meaning: clearly beats the prime number theorem) ? If so, what are the most primogenic polynomials of each degree, and just how well do they do asymptotically? --Dan A.

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