February 2005 - mathfun@mailman.xmission.com

[math-fun] (Fwd) [NOBNET 4625] Mathematical Games
by Bernie Cosell 09 May '05

09 May '05

I know that in addition to your likely being interested in this, I know that several of you have contributed to Mathematical Games over the years and so might want to get in touch with Ed... /Bernie\ ------- Forwarded message follows ------- Date: Mon, 8 Nov 2004 06:30:37 -0800 (PST) From: ed pegg <ed(a)mathpuzzle.com> Subject: [NOBNET 4625] Mathematical Games To: nobnet(a)iijnet.or.jp Martin Gardner just recently celebrated his 90th birthday. Also, maa.org is *finally* releasing the entire collection of Mathematical Games on CD rom next month. There were frightening amounts of copyright issues to work out, but they have been resolved. Catalog Code: TDG/FW04 ISBN 0-88385-545-3 List Price: $54.95 This is all 15 Mathematical Games books on a CD-Rom. I'd like to do a "behind the scenes" look at Mathematical Games for an article. Martin did lots, but he avidly used a lot of help from hundreds of people. I'd like to write an article about the background help. If you ever assisted in one of Martin's columns, I'd like to hear from you -- a note about what you did. I'll use these notes in my article. Ed Pegg Jr http://www.maa.org columnist http://www.mathpuzzle.com ------- End of forwarded message ------- -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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[math-fun] New Mersenne Prime confirmed
by Schroeppel, Richard 09 Mar '05

09 Mar '05

Tony Reix reported this morning on the NMBRTHRY list that the Mersenne prime 2 ^ 25964951 - 1 is confirmed. A triple check is expected to finsh tomorrow. George Woltman has requested that the exponent be kept off "the web" until his press release is posted, and the discoverer's local newspaper has a chance to report it. This is probably M42. It is the largest known Mersenne prime. The next smaller MP exponent known is 24036583, giving modest support to the cluster theory. The GIMPS project is burning at about 16Tflops. Every prime exponent < 15 million has been tested at least once. There are still lots of untested numbers < 25 million, so there's a chance of another, smaller, MP being found. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] New Mersenne Prime confirmed
by Phil Carmody 28 Feb '05

28 Feb '05

From: "Schroeppel, Richard" <rschroe(a)sandia.gov> > Tony Reix reported this morning on the NMBRTHRY list that > the Mersenne prime 2 ^ 25964951 - 1 is confirmed. A triple check is > expected to finsh tomorrow. > > George Woltman has requested that the exponent be kept off "the web" > until his press release is posted, and the discoverer's local > newspaper has a chance to report it. > > This is probably M42. It is the largest known Mersenne prime. The > next smaller MP exponent known is 24036583, giving modest support to the > cluster theory. With the caveat that there is no cogent cluster theory. There's no reason to believe there should be, and there's no evidence that distinguishes the sequence of Mersenne primes from one consisting of entirely independent events to any significance. Unless by "modest support" you mean the same level of support that seeing a non-black non-raven gives to the all-ravens-are-black theory. Phil ===== When inserting a CD, hold down shift to stop the AutoRun feature In the Device Manager, disable the SbcpHid device. http://www.cs.princeton.edu/~jhalderm/cd3/ __________________________________ Do you Yahoo!? Read only the mail you want - Yahoo! Mail SpamGuard. http://promotions.yahoo.com/new_mail

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[math-fun] An interesting semiprime?
by Don Reble 27 Feb '05

27 Feb '05

Mathfun'ers, Ed Pegg: Long ago, Ed Pegg sought an "interesting semiprime". I pestered him briefly, to explain what he meant by "interesting". He never explained it, but all the while, I believed that I knew what he meant: An Interesting SemiPrime (ISP) is a number which - is the product of two primes; - one has a concise proof that it's the product of two primes; and - there is scant hope of finding a proper factor. (Never mind, that "concise" is subjective and "scant hope" is temporary.) I think I found such an ISP; see http://www.graysage.com/djr/isp.txt for the rest of this message. -- Don Reble djr(a)nk.ca

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Re: [math-fun] Density of sums of squares
by POPPY9X＠aol.com 27 Feb '05

27 Feb '05

yes

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[math-fun] Unpublished photo of David Klarner, George Polya, and NG deBruijn, ca 1973
by Thane Plambeck 27 Feb '05

27 Feb '05

A few weeks ago I received a photo of these three mathematicians that was taken on the campus of Stanford University, ca 1973. There's nothing unusual about the photo (it's a standard "pose and shoot"), but I thought it might be of interest to funsters. If you click the photo at the link, a larger version will pop up. http://plambeck.org/archives/2005_01.html#000731 -- Thane Plambeck http://www.plambeck.org/ehome.htm

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RE: [math-fun] Density of sums of squares
by Schroeppel, Richard 26 Feb '05

26 Feb '05

Yes. If I remember correctly, the density falls off as 1/sqrt(logN). [RKG - is this right?] But it's easy to see the ultimate density is 0. Compute the fraction of integers with no odd-degree factor of 3, or of 7, ... . The fraction is approximately product(1 - 1/q - small change) for q = 4K+3 primes. This product goes to 0. Rich -----Original Message----- From: math-fun-bounces+rschroe=sandia.gov(a)mailman.xmission.com on behalf of David Wilson Sent: Sat 2/26/2005 1:25 PM To: math-fun Subject: [math-fun] Density of sums of squares Do the numbers of the form a^2+b^2 have density 0 with respect to Z+? -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.300 / Virus Database: 266.5.0 - Release Date: 2/25/2005 _______________________________________________ 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] Density of sums of squares
by David Wilson 26 Feb '05

26 Feb '05

Do the numbers of the form a^2+b^2 have density 0 with respect to Z+? -- No virus found in this outgoing message. Checked by AVG Anti-Virus. Version: 7.0.300 / Virus Database: 266.5.0 - Release Date: 2/25/2005

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[math-fun] negular continued fractions
by R. William Gosper 26 Feb '05

26 Feb '05

As some guy recently mentioned, regular, or "floor type" cfs of square roots of rationals are periodic palindromes, e.g., if r13:=sqrt 13, r14:=sqrt 14, 1 r13 = 3 + (r13-3 = -------------------------------------------------------------), r13-1 1 1 + (----- = ------------------------------------------------) 4 r13-2 1 1 + (----- = -----------------------------------) 3 r13-1 1 1 + (----- = ---------------------) 3 r13-3 1 1 + (----- = -------) 4 3 + r13 so sqrt 13 = 3 [1 1 1 1 6]^oo, and 1 r14 = 3 + (r14 - 3 = -------------------------------------------------------), r14 - 2 1 1 + (------- = ---------------------------------------) 5 r14 - 2 1 2 + (------- = -----------------------) 2 r14 - 3 1 1 + (------- = -------) 5 r14 + 3 so sqrt 14 = 3 [1 2 1 6]^oo. "Ceiling type", or "negular" cfs have numerators of -1 instead of 1: 1 sqrt 14 = 4 + (sqrt 14 - 4 = - ---------------------------------), sqrt 14 - 4 1 4 + (----------- = - ------------) 2 sqrt 14 + 4 so sqrt 14 = 4 {4 8}^oo, as a negular expression. But it is unusual for the negular period to be shorter than the regular one. More typically, 1 r13 = 4 + (r13 - 4 = - ---------------------------------------------------------------), r13 - 5 1 3 + (------- = - ---------------------------------------------) 3 r13 - 7 1 3 + (------- = - ---------------------------) 4 r13 - 11 12 2 + (-------- = - --------) 9 r13 + 11 r13 + 11 r13 - 13 1 -------- = 2 + (-------- = - ------------------------------------------------------------), 12 12 r13 - 13 1 2 + (-------- = - ------------------------------------------) 13 r13 - 11 1 2 + (-------- = - ------------------------) 12 r13 - 7 4 2 + (------- = - -------) 9 r13 + 7 r13 + 7 r13 - 5 1 ------- = 3 + (------- = - -------------------------------------------), 4 4 r13 - 4 1 3 + (------- = - -------------------------) 3 3 8 + (r13 - 4 = - -------) r13 + 4 so sqrt(13) = 4 {3 3 2 2 2 2 2 3 3 8}^oo. The long bursts of 2s are "wannabe 1s", precluded by the ceiling rule. We can restore 1 to the inventory of partial quotients by using numerators of -1/2 instead of -1. But this actually lengthens the period while exactly preserving the per-period information density, thus reducing it per term. Note, however, that both flavors of negular cf preserve the palindromic structure of the periods. How much shorter than a regular cf can be the period of a negular cf of the same sqrt? Crudely checking for n<36353 found regular - negular periods = 48 - 30 for n = 32148, suggesting that the question should be rephrased as some sort of ratio. The greatest odd difference in that interval was 34-23 for n = 32895 For *regular* cfs of uniformly distributed reals, P(tail<x) = lg 1+x, so about 41.5% = lg 2 - lg 3/2 of the terms are 1, yet, surprisingly, the expected value of a term (partial quotient) is infinity. But then, surprise again, the *geometric* mean of the terms of these reals is the finite constant ~2.685, named for Khinchin. This lg 1+x distribution also predicts an information density of pi^2/(6 ln 2 ln 10) ~ 1.0306 digits/term, so decimal is not such an arbitrary radix after all! Gene Salamin once computed the variance of this information density to be a constant involving zeta(3), although I once saw this disputed on a Web page I didn't undertand. Has anyone done the analogous analysis for negular cfs? In particular, do those long bursts of 2s, of expected (by me) infinite expected length, manage to render the expected term finite? Finally, all these questions repeat for the "round type" cfs, which mix numerators of 1 and -1 (or 1/2 and -1/2), the former having generally shorter periods and even higher information density than regular cfs. E.g., 1 r13 = 4 + (r13 - 4 = - ---------------------------------------), r13 - 5 1 3 + (------- = - ----------------------) 3 r13 - 3 1 2 + (------- = -------) 4 r13 + 3 so sqrt 13 = 4- (3- 2 7-)^oo (note loss of palindrome and max = 2*first), and by mere coincidence, sqrt 14 = 4- (4- 8-)^oo = 4 {4 8}^oo. --rwg PS, Macsyma Inc, seems to have revived sales of version 2.4, $500 a pop on E-Bay. They even give a bugmail eadress (which I could single-handedly inundate.)

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RE: [math-fun] Re: Bob Dylan
by Paul Reiners 26 Feb '05

26 Feb '05

Not that I know of. However in an interview in 1965 or 1966 he said that he wrote "mathematical music" or words to that effect. (I would like to give an exact reference, but I can't lay my hands on that book right now.) He did mention math and geometry in "Tombstone Blues": "The geometry of innocent flesh on the bone Causes Galileo's math book to get thrown At Delilah who sits worthlessly alone But the tears on her cheeks are from laughter" I copied and pasted those lyrics from bobdylan.com, but I always heard the first line as: "The geometry of innocence; flesh on the bone" -----Original Message----- From: Fred Kochman <kochman(a)idaccr.org> Sent: Feb 25, 2005 9:57 AM To: 'math-fun' <math-fun(a)mailman.xmission.com> Subject: RE: [math-fun] Re: Bob Dylan >Are there any songs that mention mathematicians other than Dylan's "Tangled >up in blue"?:

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