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[math-fun] puzzles, some of which are geometric
by Simon Plouffe 28 Jun '04

28 Jun '04

hello, Here is a series of puzzles : http://wpc.puzzles.com/gt-IV/uspc-04.pdf sponsored by a company (the name begins with G). They ask for a password for the document : xcode6fire Some are interesting. Simon Plouffe

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[math-fun] Knuth Vol 4
by Thane Plambeck 27 Jun '04

27 Jun '04

This is probably old news to many people, but I just noticed that this page (scroll down a bit) http://www-cs-faculty.stanford.edu/~knuth/taocp.html has quite a bit in the way of "Knuth Vol 4" preview material on it. There's easily several hundred pages of stuff there embedded in downloadable .ps.gz files. 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") * * * * * Knuth writes: The material will first appear in beta-test form as fascicles of approximately 128 pages each, issued approximately twice per year. These fascicles will represent my best attempt to write a comprehensive account, but computer science has grown to the point where I cannot hope to be an authority on all the material covered in these books. Therefore I'll need feedback from readers in order to prepare the official volumes later. The publishers have pledged to make the fascicles available in an inexpensive form, essentially at cost. (Of course, the paper and binding will probably be designed to self-destruct in a few years, so that you will have to buy the real book after it is debugged:-) If all goes as planned, Volumes 4A, 4B, and 4C will be ready in the year 2007. Some "pre-fascicles" are now available for alpha-testing: Pre-fascicle 2a (Generating all n-tuples); Pre-fascicle 2b (Generating all permutations); Pre-Fascicle 3a (Generating all combinations). Pre-Fascicle 3b (Generating all partitions). I've put them online primarily so that experts in the field can check the contents before I inflict them on a wider audience. But if you want to help debug them, please go right ahead. Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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[math-fun] sqrt(binomial) = pentanomial
by Bill Gosper 25 Jun '04

25 Jun '04

1/5 1/5 sqrt(4 - 3 ) = 3/5 4/5 3/5 2/5 2/5 4/5 1/5 1/5 - 2 3 + 3 + 2 2 3 - 2 3 + 2 --------------------------------------------------- 5 ~ 0.27161916566652. Also, I lied: 1/3 9 - 3 4 atan(----------) 1/2 4/3 4/3 1/3 4/3 5 3 (4 + 2 - 7) - 6 4 - 2 + 10 tan(----------------) = ------------------------------------------- 3 5 It isn't even trivial to *verify* this symbolically. Actually, you can just plug this into the triple angle formula and compare the two algebraic expressions. --rwg

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[math-fun] Primes in First N Digits of Pi
by Robert Baillie 23 Jun '04

23 Jun '04

These are prime: 3 31 314159 31415926535897932384626433832795028841 (first 38 digits of Pi). The first 16208 digits of Pi are a probable prime. Ed T. Prothro found this in 2001. I just finished checking up through 40000 digits. There are no other numbers in this sequence for which Mathematica's PrimeQ[ ] is TRUE. Has anyone checked beyond this limit? Bob Baillie

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Re: [math-fun] normal numbers and factorial base
by Daniel Asimov 23 Jun '04

23 Jun '04

Michael Kleber's comment << . . . (Pretending that pi is a string of random digits and other such falsities...) >> got me thinking. Suppose we have a probability distribution D on the set S of all countable sequences of decimal digits. And suppose that for all *finite* sets of indices {i_1 < ... < i_n}, the corresponding marginal distribution is uniform on {0,...,9}^n. 1. Does it then follow that D is uniform on S ??? 1a. Is there a way countably many variables can be correlated even if no finite subset of them are? Related question: I like considering numbers' "digits" in "factorial" base. For x in [0,1], I mean by this writing x = a_1/2! + a_2/3! + ... + a_n/(n+1)! + ... where each a_n is an integer such that 0 <= a_n <= n. Luckily, 1/2! + 2/3! + 3/4! ... = 1/2 + 1/3 + 1/8 + 1/30 + 1/144 +... = 1. 2. Is pi normal to the factorial base? Here I mean only the fractional part of pi. I.e., for any finite set of indices {i_1 < i_2 < ... < i_n}, does the joint distribution on a_(k + i_1)/(k + i_1+1)! , , ... , a_(k + i_n)/(k+i_n+1)! approach a uniform cdf on [0,1]^n as k -> oo ? N.B. It's easy to check that e would definitely NOT be normal to the factorial base. --Dan

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[math-fun] Double transit
by Eugene Salamin 23 Jun '04

23 Jun '04

If you look at http://iss-transit.sourceforge.net/IssVenusTransit.html you will see a photo of the International Space Station transiting Venus while Venus transits the Sun. __________________________________ Do you Yahoo!? Yahoo! Mail is new and improved - Check it out! http://promotions.yahoo.com/new_mail

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[math-fun] trig atrig
by R. William Gosper 23 Jun '04

23 Jun '04

Last week I had the pleasure to meet Albert Rich (of Derive fame), who introduced me to a class of simplifications including sqrt(15) 2 atan(--------) 9 sqrt(15) - sqrt(3) sin(----------------) = ------------------ . 3 8 With Macsyma I then concocted 1/3 9 - 3 4 atan(----------) 1/2 4/3 4/3 1/3 4/3 5 3 (4 + 2 - 7) - 6 4 - 2 + 10 tan(----------------) = ------------------------------------------- 3 5 It isn't even trivial to *verify* this symbolically. Along the way we exposed a bug in my MULTIPLY_CONJUGATES function wherein the imaginary unit, replaced by an explicit sqrt(-1), prematurely reverted to the symbol %I. --rwg

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[math-fun] a property of Bernoulli Numbers (4n+2)
by Simon Plouffe 22 Jun '04

22 Jun '04

hello, There is this puzzling property of the B(4n+2), I consider here only the fractional part, easily computable with the Von-Staudt-Clausen formula. Also, the B(4n+2) are all positives. Property ; the fractional part of the positive Bernoulli Numbers is almost never : 1/6 < {Bernoulli(4*n+2)} < 2/3. ... except for 2 cases (so far), which is {B(2072070)} = .666443506838622... and {B(6216210)} = .6588649656359250... Now, is this known? , my fist thought was that no Bernoulli(4n+2) has a fractional part between 1/6 and 2/3 , so far only 2 cases. I saw no references on this subject. The behaviour of {B(2*n)} when negative is well distributed mod 1 but the positive cases are NOT well distributed at all. Here is a graph of the succ. fractional part of Bernoulli's plotted on a circle, that is x in [-1,1] -> exp(2*Pi*x) a line is drafted between values x(0), x(1), ... http://www.lacim.uqam.ca/~plouffe/SuccFracBernoulli5000points.jpg Does someone has any reference on this? Simon Plouffe

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Re: [math-fun] a property of Bernoulli Numbers (4n+2)
by Simon Plouffe 22 Jun '04

22 Jun '04

Hello, I have found the local minimal point ( I can't explain why yet), Conjecture : there are no fractional part of positive Bernoulli numbers between 1/6 and 0.647247618323605748006387... That number 0.647247618323605748006387 is a local minimum which correspond to numbers of the form 207*10*1001*3^n that is {B(207*10*1001*3^n)} is minimal at 0.647247618323605748006387 and there are apparently no other values between 1/6 and that number. Why is it numbers of that form...I don't know. The big cluster of lines is at 1/6 of course since many B's have a fractional part of 1/6. I did the graph of {B(n)}, n=2,4,6,... to show how the fractional part behave for large n compared to a graph of succ. random lines, it is clearly not random at all. Simon Plouffe

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[math-fun] Seti : not in the top500 but completed exp(50) computations
by Simon Plouffe 21 Jun '04

21 Jun '04

Hello, the top-500 computers in the world list just came out, An interesting article about the top 500 is found here : http://www.newscientist.com/news/news.jsp?id=ns99996045 (partly by Dongarra), he talks about the fact that the Seti project is not in the top500 because of its definition. nevertheless, SETI has reached almost exactly exp(50) elementary computations that is ; 5.176068e+21 at a rate of 24.35 tera-flops (per second) which is not far from the earth similator, 35 tera-flops. http://setiathome.ssl.berkeley.edu/totals.html Simon Plouffe

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