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[math-fun] FW: New Repunit R(109297) (Harvey Dubner)
by Schroeppel, Richard 03 Apr '07

03 Apr '07

-----Original Message----- From: Number Theory List on behalf of Harvey Dubner Sent: Tue 4/3/2007 11:15 AM To: NMBRTHRY(a)LISTSERV.NODAK.EDU Subject: New Repunit R(109297) New Probable Prime Repunit, R(109297) On March 28 I found a new probable prime repunit, R(109297), which consists of 109297 1's. It is effectively impossible to prove true primality at this time. The only known repunit primes are R(2), R(19), R(23), R(317), and R(1031). In 1999 I found the PRP repunit R(49081), and in 2000 Lew Baxter found the PRP repunit R(86453). I plan to try to raise the test limit to 200000. My continued interest in repunits is due primarily to the fact that my first project on a home computer (1980) was searching for repunit primes. References: H.C, Williams and H. Dubner, The Primality of R1031, Math. Comp. 47(176), Oct 1986, 703-711. Harvey Dubner, Repunit R49081 is a Probable Prime, Math. Comp. 71(238), (2002), 833-835. A004023, On-line Encyclopedea of Integer Sequences. Harvey Dubner

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Re: [math-fun] pile of sand constant
by Henry Baker 03 Apr '07

03 Apr '07

The angle of inclination of several of the pyramids is in the range of 51.5 - 53.2 degrees. http://www.greatbuildings.com/buildings/Great_Pyramid.html Apparently, the Egyptians tried higher angles of inclination, but some of these pyramids collapsed -- e.g., Meidum. http://en.wikipedia.org/wiki/Egyptian_pyramids I would guess that the term "angle of repose" comes from the fact that these pyramids became graves? :-) :-) A pile of sand would not have the same shape _regardless of size_, as the increasing weight of the central part eventually changes the shape of the grains to cause a collapse into a smaller angle of inclination. And we in California are well aware that even relatively small amounts of water can cause spectacular collapses -- e.g., Route 1. At 09:01 AM 4/3/2007, Joshua Zucker wrote: >On 4/3/07, Simon Plouffe <simon.plouffe(a)gmail.com> wrote: >>Hello, >> >> I have this information about a certain constant >>associated with a pile of sand. Apparently the ratio >>of the half the base to the height is approx. 4/3. > >I think the geological term you are looking for is the "angle of repose". > >>I don't remember where I saw that for the first time, >>empirically, well at least for ordinary and dry sand, >>it looks like it is. In other words, a pile of sand >>will always have the same shape regardless of the size, >>or the slope is constant. > >I read that the angle of repose for dry sand is about 30 degrees, but >there are some other sites out there that tell me it depends on the >grain size. With 30 degrees, that would be sqrt(3), not 4/3. But I >think that's a pretty rough number, and I see numbers ranging up to >just under 40 degrees, which would give a ratio of about 1.2, so your >value of 4/3 seems to be in the range. The angle should be steeper >for smaller sand particles. > >--Joshua Zucker

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Re: [math-fun] pile of sand constant
by Daniel Asimov 03 Apr '07

03 Apr '07

Simon writes: << I have this information about a certain constant associated with a pile of sand. Apparently the ratio of the half the base to the height is approx. 4/3. I don't remember where I saw that for the first time, empirically, well at least for ordinary and dry sand, it looks like it is. In other words, a pile of sand will always have the same shape regardless of the size, or the slope is constant. Is there anybody that has any information about this phenomena? Is this known ? >> Thanks to the title of a noel by Wallace Stegner, I learned that the angle of the steepest pile of [whatever] is called its "angle of repose"; here it would be atan(3/4). (Cf. < http://en.wikipedia.org/wiki/Angle_of_repose >.) The angle depends heavily on the kind of stuff that is piled up. The angle of repose would be low for glass marbles, and higher for gravel of about the same size. So I doubt that the number for a pile of sand is intentionally trying to be 3/4 (or any fraction with a small denominator). --Dan

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[math-fun] pile of sand constant
by Simon Plouffe 03 Apr '07

03 Apr '07

Hello, I have this information about a certain constant associated with a pile of sand. Apparently the ratio of the half the base to the height is approx. 4/3. I don't remember where I saw that for the first time, empirically, well at least for ordinary and dry sand, it looks like it is. In other words, a pile of sand will always have the same shape regardless of the size, or the slope is constant. Is there anybody that has any information about this phenomena? Is this known ? Simon Plouffe

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[math-fun] ACRONYMS : A Common Representation Of Names You Must Shorten
by Eric Angelini 22 Mar '07

22 Mar '07

Hello SeqFans & Math-Fun, If those two seq. are of interest, could someone help in providing more terms? ------------------------ First digit of a(n) is given by the succession of digits in S [a(n+1) is the smallest available integer not yet present in S]: S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,14,31,22,15,16,41,32,17,23,24,17,51,18,61... ------------------------ First digit of a(n) is given by the succession of digits in S [S is monotonicaly increasing]: S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,111,311,2111,11111,11112,11113,11114,31111,... ------------------------ Best, É.

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[math-fun] musing with exp(Pi) , Pi and genealized continued fractions.
by Simon Plouffe 21 Mar '07

21 Mar '07

Hello, We all know that Pi and exp(Pi) have some particularities like exp(Pi)-Pi = 19.9990999... apparently found somewhere in 1988 by 3 different people... but that's not the only one, I have found this amusing continued fraction for Pi and exp(Pi), Pi - 3 = [ 1/11,-11/3, -2/12, -11/6, -5/6, -5/3, -2/6, ... ....] in continued fraction expansion. exp(Pi)-23 = [1/11, -11/3, -2/12, -11/24, -23/12, -11/24, -23/14, -13/12, -11/42,...]. There is a striking resemblance, between the two. Another one is Pi-3 = [1/11,-11^2/34, -23^2/172, -149^2/2176, ...] exp(Pi)-23= [1/11,-11^2/34, -23^2/186, -163^2/16042,...] This output comes from an algorithm I developed to <force> prime numbers in the numerator of the gen. continued fraction of some numbers. I computed the expansion of many numbers and these 4 examples have a kind of interesting pattern. Simon Plouffe

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[math-fun] (doubtless) old segment trick
by R. William Gosper 20 Mar '07

20 Mar '07

Given pts A,B in n-space, instead of representing the segment AB by tA + (1-t)B and later checking 0<t<1, just use uA + B, scaling whatever gets added by u+1, and later check u>0. E.g., to see if AB intersects the plane thru P with normal N, solve for u N.(uA+B)=(u+1)N.P, where N.P can be precomputed. The actual intersection is (uA+B)/(u+1). --rwg GREAT PLANES PLANET GEARS PS, apropos Jeopardy!, the TV at Deedees Vegetarian Buffet last nite showed Crorepati, an exact homologue of Philbin's Millionaire show (perhaps minus the lifelines). But winnings were displayed as Rs x,xx,xxxxx instead of $ x,xxx,xxx. (And the quizmaster can break into song.) The math question seemed easy enough: Which of the following is greater than 86? <Four strange words ending in "asi".> Oops.

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[math-fun] Atlas of Finite Groups query
by Fred lunnon 20 Mar '07

20 Mar '07

The Atlas of Finite Groups at http://brauer.maths.qmul.ac.uk/Atlas/ has been the subject of my recent (somewhat apprehensive) investigations. As expected, it proves to make few concessions to the neophyte. In particular, I would be grateful if some kind group-theorist could enlighten me as to the meaning of "19:9" and similar (modulo typo) expressions, occurring frequently therein, but nowhere defined! Fred Lunnon

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[math-fun] E8 : lie group breakthrough
by Simon Plouffe 20 Mar '07

20 Mar '07

Hello, this just came out, http://news.google.com/news?hl=en&q=e8&btnG=Google+Search&ie=UTF-8&oe=UTF-8… it appears in a number of sites already. What is it that we could not do up to now in fact? Simon Plouffe

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[math-fun] Meta analysis spoof
by Henry Baker 19 Mar '07

19 Mar '07

FYI -- If you haven't seen or heard of this paper, it is hilarious... Parachute use to prevent death and major trauma related to gravitational challenge: systematic review of randomised controlled trials http://www.bmj.com/cgi/content/full/327/7429/1459

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