June 2004 - mathfun@mailman.xmission.com

[math-fun] Twin prime proof offered
by Richard Schroeppel 09 Jun '04

09 Jun '04

Title: There Are Infinitely Many Prime Twins Authors: R. F. Arenstorf http://arXiv.org/abs/math/0405509 http://www.math.vanderbilt.edu/faculty/Arenstorf.html Richard Arenstorf is a real mathematician, at Vanderbilt. I looked at the 38 page manuscript - it's a well presented bunch of analytic number theory. The proof requires a lot of multi-level series manipulations, and careful arguments about convergence. I'm way too rusty to check this, but it's clearly a serious attempt. He notes that he's been working on the problem intermittently for 20 years. A Google search turned up a few references to his other work. The author claims a bit more: That the expected asymptotic expression for the count is correct, including the twin-prime constant. My quick scan didn't show any reason the proof wouldn't adapt to more general {P,P+2k} twins, although "Twin" would have to be agnostic about the primality of the intermediate numbers between P+2 and P+2k-2 when k>2. Similar ideas might also show the infinity of Sophie Germain pairs P,Q=2P+1 with P and Q both prime; and a restriction to P=3(mod4) would show an infinity of Mersenne composites Mp. The more general problem of primes in arithmetic progressions (longer than 2 of course) was recently settled by a density argument. Arbitrary lengths exist. I'd welcome comments from someone who remembers more of their analytic number theory than I do. Rich rcs(a)cs.arizona.edu

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Re: [math-fun] Permuation Probabilities ... with apologies ...
by Daniel Asimov 09 Jun '04

09 Jun '04

Don reble wrote: << [Guy Haworth wrote: > In the 'knockout' first round involving 128 players and 64 matches, what > is the probability that none of 19 specific players are matched against > each other? 126*124*122*120*...*92 Could it be ---------------------- ~= .20822833 ? 127*126*125*124*...110 >> I wrote what amounts to << 109*108*107*...92*91 --------------------, 127*125*123*...93*91 >> which is ~= .20822833 also, so I presume Don's answer is another way to calculate the same probability. But I don't see why yet. Can someone please explain his method to me? --Dan

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[math-fun] Abramowitz & Stegun book in acrobat PDF,
by Simon Plouffe 08 Jun '04

08 Jun '04

Hello, apparently that book is widely distributed now these days. Unfortunately the copy we see in a couple of places is of poor quality. I made one myself a while ago with my own copy that I did cut and pass through the scanner. http://www.lacim.uqam.ca/%7Eplouffe/articles/Abramowitz&Stegun.pdf It is in the PDF format and of course I made the OCR on it. The result is not perfect but it is better than the one we see in other places. ps : I do not think that this is an infrigement of the copyright because the document is available in some place of the net, if ever it is I would remove it and of course the one I give away like this is NOT for sale, it is for research only. Simon Plouffe

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[math-fun] Permuation Probabilities ... with apologies ...
by Guy Haworth 07 Jun '04

07 Jun '04

.. for the dumb question, but I could use the answer faster than I can work it out (re the upcoming FIDE World Chess Championship). In the 'knockout' first round involving 128 players and 64 matches, what is the probability that none of 19 specific players are matched against each other? Not quite the 'birthday problem' but ... Guy

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RE: [math-fun] Permuation Probabilities ... with apologies ...
by Daniel Asimov 07 Jun '04

07 Jun '04

Guy Haworth asks: << re the upcoming FIDE World Chess Championship). In the 'knockout' first round involving 128 players and 64 matches, what is the probability that none of 19 specific players are matched against each other? >> If we assume the first 19 assignments of first-round opponent are to these 19 players, the probability you request must be ((128-19)/128-1) * (128-19-1)/(128-3))* ... *((128-19-18)/(128-37)) since the first player must be assigned to any of 128-19 out of 128-1; the second must now be assigned to any of 128-(19+1) out of 128-3; until the last of the 19 must be assigned to any of 128-(19+18) out of 128-36, which is all that matters. --Dan

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[math-fun] Fifty year anniversaries
by Thane Plambeck 07 Jun '04

07 Jun '04

Under what circumstances exactly will a 50 year anniversary fall on the same day of the week as the original date? Today (6 June 2004) is a Sunday, and so was 6 June 1954. Ignore all the old calendar stuff, let's say I'm only interested in this question for dates after 1800. There seem to be lots of web pages about "repeating calendar years" but that's not at all the same question since dates can agree on day of the week even when they differ on leap year status... I was asked this at a wedding anniversary party and stammered something about 100 year leap year cycles with 400 year exceptions but couldn't work out the general answer, at least with dance music playing Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] Does 1=1?
by Jon Perry 06 Jun '04

06 Jun '04

If so, a lot of contemporary mathematics would flow easily... But jokes aside, what is the concept that binds my 1 to your 1? 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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[math-fun] when is a certifiable prime 'surprisingly easy to prove'?
by wouter meeussen 05 Jun '04

05 Jun '04

the elliptic curve method in Mathematica's (NumberTheory`PrimeQ`) package says after 20 seconds that 15085130035827878542455979623747888891433345604817588712723282399687865427853871 is prime. An 80 digit number. Should I expect such timing, or does this large integer have special properties that make it easy to get a certificate? W. quote When \!\(TraditionalForm\`n\) is larger than the value of the option SmallPrime, ProvablePrimeQ uses the Atkin\[Hyphen]Morain test as described above. This primality proving algorithm is suboptimal if the number is within the range of efficient factoring algorithms. When this is the case, a method of V.\[NonBreakingSpace]Pratt is preferable: V.\[NonBreakingSpace]Pratt, \[OpenCurlyDoubleQuote]Every Prime Has a Succinct Certificate,\[CloseCurlyDoubleQuote] SIAM Journal of Computation 4 (1975), pages 214\[Dash]220. An implementation of this algorithm in Mathematica is described in S. Wagon, Mathematica in Action, W. H. Freeman, 1991, Section\[NonBreakingSpace]8.7. end quote

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[math-fun] abcdef conjecture
by Jon Perry 04 Jun '04

04 Jun '04

I've not seen this anywhere else, can we generalize the abc conjecture into something whereby; f=a+b+c+d+e or any number of variables, and a form of the original abc conjecture still exists? I am aware of a conjecture that for the abc conjecture there is an upper bound of zeta(2): http://www.mathematik.uni-jena.de/~aros/abc.html Do other zeta(x) limits exist for the general abc conjecture? 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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[math-fun] odometer word game
by Richard Schroeppel 03 Jun '04

03 Jun '04

zimb zinc chromatic chromatid Rich

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