August 2004 - mathfun@mailman.xmission.com

RE: [math-fun] Re: More innumeracy in high places
by Andy Latto 09 Aug '04

09 Aug '04

> > >Third, give the proof. It would make > >for an interesting experiment to see what finally appears in print. > > Probably something like this: > > "The proof is based on the method of self-contradiction, > in which an assertion is assumed to be both true and false > at the same time." I'm sure the press could find some other way to garble the proof, but this particular garbling could be avoided by using a direct, constructive, proof, rather than a completely unnecessary proof by contradiction. Theorem: For any N, there is a prime > N. Proof: Consider any prime factor of N!+1 What does the traditional insertion of "first suppose there are no primes > N" add to this proof, other than the possibility of confusion? Andy Latto andy.latto(a)pobox.com >

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[math-fun] Re: More innumeracy in high places
by James Propp 09 Aug '04

09 Aug '04

>Third, give the proof. It would make >for an interesting experiment to see what finally appears in print. Probably something like this: "The proof is based on the method of self-contradiction, in which an assertion is assumed to be both true and false at the same time." :-( Jim Propp

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Hartmut Holzwart/IS-F1 A1/Allianz-Leben ist außer Haus. : [math-fun] Getting a complete set
by hartmut.holzwart＠allianz.de 08 Aug '04

08 Aug '04

Ich werde außer Haus sein vom 03.08.2004 bis 16.08.2004. Bitte wenden Sie sich an Herrn Peter Schumann (allgemein), Herrn Frank Tiemann (APE, Compliance, CL, SCD) bzw. Herrn Marco Di Doi (WEPA-Themen), Herrn Christoph Weiß (SOX) Ursprüngliches Thema: [math-fun] Getting a complete set

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[math-fun] More innumeracy in high places
by David Gale 08 Aug '04

08 Aug '04

The quote below appeared in a column in the International Herald Tribune by Michael Johnson described as "a former correspondent for Business Week and The Associated Press." The column head was "2,3,5,7,1,13,,19,23 What comes next?" The article was about the search for a ten million digit prime. About halfway down we find, "It was Euclid who started it all by identifying the smaller primes 2,300 years ago and predicting that the number of primes would turn out to be infinite [my italics] ." I don't know, wouldn't you think even the man or woman on the street should be able to digest Euclid's proof? I'm tempted to go out on the street and try to find out. "Look, Mary, suppose there were only a zillion of them. Well then, multiply them all together and add 1and you see . . . " What do you all think? David G

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Re: [math-fun] Getting a complete set
by Phil Carmody 07 Aug '04

07 Aug '04

From: Chris Landauer <cal(a)rush.aero.org> > hihi, all - > > i thought about this same problem some years ago, and will try to recreate > where i ended - the original problem was to estimate the size of a picture > archive on the web, from the repetition (presumed independent) of single > experiment that extracted (presumably uniformly randomly) and presented one > picture > > the question was when to stop looking at the set, so i wanted a reasonable > estimate for the total size N of the set ... > i tried to find the problem statement in some published application, and the > closest i could get was the population estimation problem in ecological > sampling This is simply the birthday paradox (it's not a logical paradox, it merely provides some initially non-intuitive numerical results). Given the historical use of birthday attacks in mathematics and cryptography (for factoring, discrete logarithms, et al.) I'd hope there would be a reasonably body of literature on the subject. Your Bayesian approach doesn't deviate so much from the factoring and discrete logarithm uses of the technique, as there one doesn't know the "size" of the pseudo-random permutation /a priori/. 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!? Yahoo! Mail Address AutoComplete - You start. We finish. http://promotions.yahoo.com/new_mail

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Re: [math-fun] More innumeracy in high places
by Phil Carmody 07 Aug '04

07 Aug '04

From: Thane Plambeck <thane(a)best.com> > > At the ams.org, and many other places (including GIMPS), the > phrase "The Largest Known Prime" is used repeatedly. There > must be some other way to report Mersenne prime progress without > using this unfortunate phrase, which finds itself into the media and can be > easily misconstrued as "Largest Prime Found." > > I don't have a good replacement though. Maybe just "Larger" instead? But the "Largest Known Prime" does mean the "Largest Prime Found". There is no misconstrual. To what do you want the phrase "Largest Known Prime" to refer, if not the largest prime found? And what to you mean by the "Mersenne prime progress" anyway? All the knowledge in the world about composite Mersennes (which is what most of the information, by count, coming out of GIMPS is) does not affect in any way the knowledge of what the largest prime found is. However, the idea that somehow Mersenne prime testing progress is tied to the worlds largest prime is conveniently forgetting Riesel and the Amdahl 6, and also ignoring the fact that there are other current projects which would have a more favourable chance of finding a prime larger than the current largest known one, if only they would have the popularity of GIMPS. For example, Yves Gallot's Generalised Fermat Number (GFN) hunt is looking at ranges of numbers all of which have a similar size (this helps keep the density roughly constant), has an exceptionally simple right-angle DWT transform for fast modular arithmetic (simpler than Mersenne's IBDWT, basically the same work-factor), can be sieved rather than simply trial-divided, and which have an exceptionally high prime density (Mersennes are sparse due to the primality criterion on the exponent). If it had the same CPU resources as GIMPS it's yield would be roughly twice GIMPS's (as measured by the sum of the cubes of the logs, extrapolating the current yield from their limited resources). My own PIES project looks at ranges of Generalised Eisenstein Fermat numbers (the Phi(3*2^n,b) equivalents to GFN's Phi(2^n,b), corresponding to norms of Eisenstein integers b^(2^n)+omega rather than GFN's norms of b^(2^n)+i in the complex plane) which have the same similar-size property as Yves' GFNs, has a dogleg-DWT (my own coinage, I've not seen anyone use such a transform before, but it has the same work-factor as an IBDWT and a right-angle-DWT), can also be sieved, and has _twice_ the density of Yves' GFNs, thus having a higher chance of finding results in the finite ranges (bounded by FFT rounding errors) available. (It is the 4-th highest yielding project despite only having a handful of testers with a handful of machines, many of which are obsolete technology.) The days of the Mersenne's dominance because of the 3x work-factor boost from its IBDWT is now over. It's now only dominant as it has unimaginably more boxes working on it. 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!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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[math-fun] google statistics
by Thane Plambeck 07 Aug '04

07 Aug '04

"largest prime number found" --- 231 results "larger prime number found" --- 0 results "largest known prime number" --- 3890 results "larger known prime number" --- 0 results We live in a superlative society. -- Thane Plambeck thane(a)best.com http://www.plambeck.org

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Re: [math-fun] Critique of the Monthly
by Daniel Asimov 06 Aug '04

06 Aug '04

Gene writes << << [Here Gene quotes my expressed wish that an "introduction to QM" article in the Monthly had been written in standard math -- not physics -- notation.] >> But this contradicts your earlier request for using standard notation. The standard notation for quantum mechanics is that which is used by the physicists. Perhaps what is really inappropriate about this paper is attempting to explain quantum mechanics within the very brief space permitted. It would be a better allocation of resources to cite some standard textbooks, and proceed to the main point of the article. >> You are so right. But I meant math notation. Regrets if that was not clear from the context. > ... << One of my main gripes about the Monthly is that sometimes the articles are dumbed down. Some years ago there was a paper about one dimensional manifolds. Great, I thought, maybe I'll learn something about the long line. But no, the author acknowledged that there was one more manifold called the long line, but it was outside the scope of the the paper. Eventually I decided that the MAA and the Monthly were outside my scope, and allowed my membership to lapse. >> Since the time I was in high school, at least, its aim used to be accessible to undergraduates. For the last 15 or so years, however, it looks to me to be aimed at 2nd-year math grad students and up, or rather advanced undergrads. The change was probably gradual. I find it much more advanced than it used to be years ago. (Though it probably still runs a few "dumbed down" articles just to attract more undergrads, these are not typical.)

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[math-fun] Heron triangles
by Bill Gosper 06 Aug '04

06 Aug '04

Is it known that 1 and 2 are the only impossible sides of (integer) Heron triangles? It appears that all such triangles with side 3 are of the form 3 n n 1 [- ((sqrt(8) + 3) + (3 - sqrt(8)) ) + -, 4 2 3 n n 1 - ((sqrt(8) + 3) + (3 - sqrt(8)) ) - -, 3], 4 2 and all with side 6 are either twice that, or 3 fib(6 n) 3 fib(6 n) [3 fib(6 n - 1) + ---------- + 2, 3 fib(6 n - 1) + ---------- - 2, 6], 2 2 or the lone isosceles case [5,5,6] (= twinned [3,4,5]s). Similarly for a side of any small integer. --rwg radiolytic cordiality

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Re: [math-fun] Knuth Vol 4
by Michael D Beeler 06 Aug '04

06 Aug '04

Thane Plambeck wrote on 06/27/2004, about "Knuth Vol 4" preview material: Here's a puzzle from it [rated 25 on the Knuth scale] What is the 7.2.1.2.26: What is the unique alphametic solution to NIIHAU \pm KAUAI \pm OAHU \pm MOLOKAI \pm LANAI \pm MAUI \pm HAWAII = 0? (\pm : "plus or minus") This is an especially nice puzzle, because there is also a (different) unique answer if the "= zero" is mis-read as "= the letter O". -- Mike

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