June 2009 - mathfun@mailman.xmission.com

[math-fun] formula for the catalan constant, eisenstein series reformulated and other findings.
by Simon Plouffe 14 Jun '09

14 Jun '09

hello, I just found this formula for Catalan constant , is this known ? /infinity \ /infinity \ | ----- | | ----- | | \ 1 | | \ 1 | 11 | ) -------------------| - 71/2 | ) ---------------------| | / 2 | | / 2 | | ----- n (cosh(Pi n) - 1)| | ----- n (cosh(2 Pi n) - 1)| \ n = 1 / \ n = 1 / /infinity \ | ----- | | \ 1 | + 11 | ) ---------------------| | / 2 | | ----- n (cosh(4 Pi n) - 1)| \ n = 1 / gives : 11*Sum(1/n^2/(cosh(Pi*n)-1),n = 1 .. infinity)-71/2*Sum(1/n^2/(cosh(2*Pi*n)-1), n = 1 .. infinity)+11*Sum(1/n^2/(cosh(4*Pi*n)-1),n = 1 .. infinity) when lprinted and evalf(%) gives : 0.91596559417721901505460351493238411077414937428167213426649811962176301977\ 625476947935651292611510624857442261919619957903589881... I think it is valid, one thing I know : I never saw it before. if I am not mistaking, that formula can be used to compute catalan constant to a high precision. I have others that are of interest here : http://pictor.math.uqam.ca/~plouffe/inspired3.pdf a html version is available also at http://pictor.math.uqam.ca/~plouffe/ see the middle of the page some of the text is in french but formulas are in universal language with <sigma> of course, this is a draft of course. some findings are related to Eisenstein series, that I find simpler to grasp in some way. ah yes, I have also found a generalization of Ramanujan formula for rational arguments, as you will see in the documents, instead of having only the cases with 4n+1 there is now a general formula for all odd index as you will see. bonne lecture, simon plouffe

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Re: [math-fun] The hexadeciaml pi formula; Catalan &al.
by Dan Asimov 14 Jun '09

14 Jun '09

By the way, my understanding is that Simon Plouffe is 100% responsible for the so-called BBP formula. >From Math Reviews: << MR1479991 (98h:11166) 11Y60 Adamchik,Victor [Adamchik,V. S.]; Wagon, Stan (1-MACA) Asimple formula for . Amer.Math. Monthly 104 (1997), no. 9, 852–855. The authors show how to discover and prove formulas like the recent formula for given by D. H. Bailey, P. B. Borwein and S. Plouffe [Math. Comp. 66 (1997), no. 218, 903–913; MR1415794 (98d:11165)] which allows the computation of the nth base-16 digit of without knowing the previous digits. They use Mathematica to find a different but similar formula. Specifically, they use Mathematica to sum the series [a similar family of formulas goes here] and then determine the coefficients so that the sum is They then show how to give an integral equivalent formulation that eases the task of producing a verifiable proof. --Dan Rich wrote: << Simon Plouffe was one of the discoverers of the nice pi formula inf 4 2 1 1 -k pi = sum (---- - ---- - ---- - ----) 16 k=0 8k+1 8k+4 8k+5 8k+6 which allows computing individual bits of pi without computing the preceding part. It seems to me that there should be another such formula, in which the mod 8 residues are rearranged a little. Perhaps with 8k+3 and 8k+7 replacing 1 & 5, or maybe just 1&3 instead of 1&5, or with 8k+2 in place of 8k+6. Has anyone seen variations like this? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Fwd: hejhal's tribute to atle selberg
by rcs＠xmission.com 13 Jun '09

13 Jun '09

This is mostly personal reminiscences, with some math flavoring. Rich ----- Forwarded message from rjbaillie(a)frii.com ----- Date: Sat, 13 Jun 2009 11:57:06 -0400 From: Robert Baillie <rjbaillie(a)frii.com> Reply-To: Robert Baillie <rjbaillie(a)frii.com> Subject: hejhal's tribute to atle selberg To: Richard Schroeppel <rcs(a)xmission.com> hi rich, i don't know if you saw this or not, but: http://www.ams.org/notices/200906/ bob ----- End forwarded message -----

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[math-fun] new mersenne prime found, but ...
by Robert Baillie 12 Jun '09

12 Jun '09

it's the second largest now known, not a new record. it's 2^42643801 - 1 see http://www.mersenne.org/ for more details

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[math-fun] An ear for idiom
by Hilarie Orman 11 Jun '09

11 Jun '09

Heather McHugh's poem "Hackers Can Sidejack Cookies" in the May 11 New Yorker is a real surprise. Although it is a self-described collage of hacker jargon, the juxtaposition of the phrases echo conversations from decades of my own hacker life, and the arrangement is oddly pleasing. "Who wrote these tunes, these runes you need black art to parse?" That verse could apply to mathematics as well, and I suppose McHugh could have taken Niven and Zuckerman's elementary number theory book for the starting point of a collage. But "quadratic reciprocity" cannot compare to the imagery of "A berserker wizard gets no score for treasure." About 20 years ago the New Yorker published a short story based math word problems. I think it started, "A lies in bed next to B, dreaming of C." It seems to me that there have been few attempts to work the language of sci-math-tech into literature. Maybe McHugh's poem will inspire others to use nerdspeak beautifully in other formal writing, as if it were any other kind of valued ethnic speech pattern. McHugh was near MIT back in the day, I wonder if she ever had occasion to visit MAC or TMRC. Hilarie

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Re: [math-fun] Bernoulli numbers
by N. J. A. Sloane 09 Jun '09

09 Jun '09

A later news item reports that the formula is well-known. http://www.uu.se/news/news_item.php?typ=artikel&id=693 Neil

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[math-fun] Descriptive Sudoku
by Ed Pegg Jr 07 Jun '09

07 Jun '09

Here's a hard one I created: a descriptive sudoku. The solution is unique. In a Jigsaw sudoku, numbers 1-9 occur once in each row, column and irregular 9-square region. One jigsaw sudoku has three non-touching 3x3 squares, each filled with the digits 1-9 in order, and each touching 2 or 3 of the other six *identically-shaped* regions. One 3x3 square is in the upper left corner, and another is high up on the right side. The main diagonal has four 1's. As a hint, the identical shape used in the six regions is a 2x4 rectangle with a square added to one corner. Ed Pegg Jr

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[math-fun] Determinant Sudoku
by Jason 07 Jun '09

07 Jun '09

Hilarie and Rich tell me they have trouble finding sufficiently difficult sudoku puzzles to keep them occupied. So one day while daydreaming I came up with Determinant Sudoku, in which you take the determinant of each square within a valid sudoku and make a super-square in which no row or column repeats a determinant. In base-4 sudoku, such a Determinant Sudoku is a grid of 12x12 numbers, not too much larger than a base-10 ordinary sudoku. But I wasn't sure if any valid base-4 Determinant Sudoku actually existed, so I wrote a program to help me check. I did manage to construct a valid base-4 Determinant Sudoku, but I'm not sure how many others exist. Here is the source code to enumerate all base-4 sudoku and print their determinants: http://sites.google.com/site/credentiality/Home/determinant-sudoku.c?attred… And I've copied the comment header below which includes more explanation and the base-4 determinant sudoku I found: // This program enumerates all the base-4 sudoku and calculates // their determinants. // // The idea is this: start with base-4 sudoku, so that you get // 2x2 squares which have the numbers 0,1,2,3 with none repeated. // Create a 2x2 grid of those 2x2 squares, so that each row and column // has the numbers 0,1,2,3. (Ordinary sudoku, but 4x4 instead of 9x9). // // Here's an example of a valid base-4 sudoku: // 01 23 // 23 01 // // 10 32 // 32 10 // // There appear to be 288 distint base-4 sudoku, without removing isomorphisms. // // My idea was to take the determinant of each of those 2x2 matrices and create // a "determinant sudoku". // // In matrix math, the determinant of // a b // c d // is a*d - b*c. // // There are six possible determinants of the 2x2 blocks. // // To construct a base-4 determinant-sudoku, make a 3x3 grid of 4x4 base-4 // sudoku blocks. There will be six rows and six columns of 2x2 squares. // // How many base-4 determinant-sudoku blocks exist? // This program almost answers that question, but not quite. // // Of the 288 base-4 sudoku, 168 have the same determinant, leaving 120 // base-4 sudoku that could contribute to a base-4 determinant-sudoku. // // I constructed one base-4 determinant-sudoku by hand: // // 01 23 | 01 32 | 02 31 // 23 01 | 32 01 | 31 02 // | | // 10 32 | 10 23 | 20 13 // 32 10 | 23 10 | 13 20 // // ---------------------- // // 01 32 | 02 31 | 01 23 // 32 01 | 31 02 | 23 01 // | | // 10 23 | 20 13 | 10 32 // 23 10 | 13 20 | 32 10 // // ---------------------- // // 02 31 | 01 23 | 01 32 // 31 02 | 23 01 | 32 01 // | | // 20 13 | 10 32 | 10 23 // 13 20 | 32 10 | 23 10 // // // Determinants: // // -2 2 -3 3 -6 6 // 2 -2 3 -3 6 -6 // // -3 3 -6 6 -2 2 // 3 -3 6 -6 2 -2 // // -6 6 -2 2 -3 3 // 6 -6 2 -2 3 -3 // // Each 4x4 block is a valid base-4 sudoku. And the 6x6 grid of 2x2 blocks // has no repeated determinants in any row or column. // // (This base-4 determinant sudoku isn't very nice, since it repeats the // same 3 base-4 sudoku 3 times each.) // // So the question remains open: how many base-4 determinant sudoku are there?

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Re: [math-fun] An ear for idiom / Bernoulli numbers
by Dan Asimov 01 Jun '09

01 Jun '09

Quel surprise! --Dan << At least one news source is reporting that the Altoumaimi's formula is not new: http://news.yahoo.com/s/afp/20090528/od_afp/swedeneducationoffbeatreply_200… >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] An ear for idiom / Bernoulli numbers
by N. J. A. Sloane 01 Jun '09

01 Jun '09

> Heather McHugh's poem "Hackers Can Sidejack Cookies" in the May 11 New Yorker is a real surprise. I too liked that poem a lot, especially the line and wetware lives in the meatspace ------- Changing the subject, has anyone seen the formula referred to in this news item? Iraq-born teen cracks maths puzzle STOCKHOLM (AFP) A 16-year-old Iraqi immigrant living in Sweden has cracked a maths puzzle that has stumped experts for more than 300 years, Swedish media reported on Thursday. In just four months, Mohamed Altoumaimi has found a formula to explain and simplify the so-called Bernoulli numbers, a sequence of calculations named after the 17th century Swiss mathematician Jacob Bernoulli, the Dagens Nyheter daily said. Altoumaimi, who came to Sweden six years ago, said teachers at his high school in Falun, central Sweden were not convinced about his work at first. "When I first showed it to my teachers, none of them thought the formula I had written down really worked," Altoumaimi told the Falu Kuriren newspaper. He then got in touch with professors at Uppsala University, one of Sweden's top institutions, to ask them to check his work. After going through his notebooks, the professors found his work was indeed correct and offered him a place in Uppsala. Neil

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