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[math-fun] Primes in First N Digits of Pi
by Phil Carmody 09 Jul '04

09 Jul '04

Recently (before I was subscribed) Robert Baillie mentioned that he'd used Mathematica to search for primes in the leading digits of pi, to 40000 digits. For this size of number, the fastest program for probable primality testing is Chris Nash, Jim Fougeron et al.'s OpenPFGW, which uses George Woltmann's FFT library (as used in GIMPS) for its incredible speed. Similarly the most efficient way to avoid testing is to use some kind of sieving technique to effectively perform trial-division on the candidates and reject the obvious composites. To that end, I've written a trial-division-sieve (not a true sieve, but not naive trial division either, somewhere in between in efficiency), and run it over a range of candidates from 40000 to 100000 digits. I reached a depth approximately 30 times the depth that OpenPFGW's own naive trial division performs. This, using Mertens' theorem, implies that there are about 15% fewer candidates to now test than such trial division. I have no idea how deep Mathematica balances trial-division and PRP testing, but I suspect that it is incomparably less efficient than PFGW at these sizes. I have no spare CPU power to test these numbers myself, but if interested parties want to contact me, I can let them have the presieved ranges, and a PFGW script file that can test the range. Robert can have dibbs, of course. I did some simple back-of-a-fag-packet calculations, and I suspect that the range contains ~0.4 primes, so it will be a stroke of luck if one is found. However, if one is found, I suspect that it will remain the largest such PRP for quite a long time, as the expectation between 100000 and 250000 digits is also only 0.4 (as was the expectation between 16000 and 40000) My TDSieve can be trivially adapted to other constants (e, phi, sqrt2, etc.) and to other bases, so if others are interested in related tasks, please don't hesitate to contact me regarding them (or keep it on-list). 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 - 50x more storage than other providers! http://promotions.yahoo.com/new_mail

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[math-fun] Re: the number 7088 is of no particular interest.
by Robert G. Wilson v 08 Jul '04

08 Jul '04

Et al, Will I be able to prove that 2+2 = 5 using the logic below? Seriously, how close to the Benford Law does the OEIS follow? Bob. Simon Plouffe wrote: > > > Hello, > > (this is a useless statistic) > > in the old days the number 417, did not appear > anywhere in the EIS (as of 1994) just before > the publication of the book, now the > > smallest integer not being in the table is 7088. > > Which means that this number is of small interest > (so far). > > The statistic is also of small interest. > > note : 7088 = 2^4 * 443 > > Obviously all numbers are included in A000027 > (the natural integers). > > this is surprising that 2^4 * 443 is not there, isn't ? > > Here is a graph of the log(# entries) from 1 to 30000 > > http://www.lacim.uqam.ca/~plouffe/OEISstatistics.gif > > Simon Plouffe >

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[math-fun] Re: Even Jeb Bush is smarter than the NYT
by Dean Hickerson 08 Jul '04

08 Jul '04

Mostly to Eugene Salamin: > I don't fault Jeb for not knowing the answer. Governors are not > usually elected for their mathematical skills. ... > You seem to have missed the essential point of my message. It was not > an attack on Gov. Bush, or on an 18-year old student, or even public > education. It was an attack on the New York Times. Newspaper reporters aren't selected for their math skills, either. So why do you consider it OK for a governor not to know, but you consider it bad that a reporter didn't know? As usual, your own biases determined the conclusion that you drew from the story. Dean Hickerson dean(a)math.ucdavis.edu

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RE: [math-fun] perfect squares
by Schroeppel, Richard 07 Jul '04

07 Jul '04

If a+b*sqrtK is the square of, say, c+d*sqrtK, then a-b*sqrtK is the square of c-d*sqrtK. So a pretest is to see if the product a^2 - K b^2 is an integer or rational square. Assuming this condition is passed, then 2c = sqrt(a+b*sqrtK) +- sqrt(a-b*sqrtK). In the integer case, this calculation can be done in floating point, and generates two possible candidate c values. C must be an integer or integer+1/2. The corresponding d value must meet a = c^2 - K d^2 and b = 2cd. One equation gives candidate d's, and the other checks them. For the rational case, either deal with it directly (since the denominator of c must divide 2 * den(a) * den(b), the floating point c value can be multiplied by 2 den(a) den(b) and must be an integer) or convert the initial problem to integers by multiplying by (den(a)den(b))^2. I'm not sure if this qualifies as "easy", and it's got some annoying branches in the algorithm. Rich -----Original Message----- From: Bill & Rosemary Cordwell To: math-fun Sent: 7/7/2004 5:16 PM Subject: [math-fun] perfect squares Is there an easy way to tell when a + b*sqrt(k) is a perfect square in the extension field Q(sqrt(k))? For example, 3 + 2*sqrt(2) is the square of 1 + sqrt(2). I'm particularly interested in when a, b, k are integers, but a and b rationals would be OK, too. Thanks, Kristin Cordwell _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] atan(3/4)
by Schroeppel, Richard 07 Jul '04

07 Jul '04

The smallest angle of a 3-4-5 triangle is 36.87 degrees = atan(3/4). One way to get a decent approximation is as atan(1) - atan(1/7). The first term is 45 degrees, and the second is fairly approximated as 1/7 radians, which is 57.3/7 = ~ 8.2 degrees. This gives 36.8 degrees as an approximate answer. The first neglected term in the atan power series is 1/3*7^3 = 1/1029, so the approximation is decent. The nearness to 36 degrees makes the 3-4-5 triangle helpful in drawing approximate regular pentagons on graph paper. Rich rcs(a)cs.arizona.edu PS: Let's try to avoid political discussions. We've all been through the arguments about the quality of math education and related topics. Unless you have something truly new to say on the subject, there's no benefit to reiteration on this mailing list.

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Re: [math-fun] Even Jeb Bush is smarter than the NYT
by Jeffrey Shallit 07 Jul '04

07 Jul '04

> Lest anyone be under the illusion that the New York Times has even the > slightest degree of credibility in their scientific articles, I quote > from the July 7, 2004 issue, page A17. I find it remarkably silly to indict the scientific coverage of the NYT by quoting from a single AP article. Surely any intelligent assessment of the quality of this coverage should consist of more than one data point. Compared to my local newspaper, the NYT coverage is postively stellar. My local paper routinely publishes articles in favor of dowsing, chiropractic, psychic phenomena, etc.

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[math-fun] sqrt(binomial)=octonomial, sort of
by R. William Gosper 07 Jul '04

07 Jul '04

I said >Still no luck with sqrt(binomial) = hexanomial. 1/4 sqrt(113 %i - 84 2 ) = 3/4 1/4 3 %i 3/4 1/4 3 - 2 %i + 4 2 %i + ------- + 6 %i + 2 - 4 2 + ------- + 6, sqrt(2) sqrt(2) but half of this is from sqrt(i)=(i+1)/sqrt(2). --rwg

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[math-fun] Least dense packings
by Jon Perry 07 Jul '04

07 Jul '04

Let C be a 2d container with height h and width w from the non-negative reals. Unit circles (radius 1) can be placed inside the container, and they obey the usual rules of gravity, and do not intersect. What is the minumum number of unit circles needed to place a circle such that its highest point is of height >= h? If w<2 then f(h,w)=oo as we cannot place any unit circles into a container of this width. At w=2, f(h,w)=ceil(h/2) but for w>2 f becomes much less defined. 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] trig denesting
by R. William Gosper 05 Jul '04

05 Jul '04

Albert Rich points out that you get simplification when the complex cube root denests in 2 1/6 1/3 atan(x) (x + 1) (%i - x) sin(-------) = ------------- + ------------- . 3 1/3 2 1/6 2 (%i - x) 2 (x + 1) This seems to happen fairly rarely, but a nice one is 1/3 4 + 1 atan(--------) 1/3 sqrt(3) 4 - 1 tan(--------------) = --------, 3 sqrt(3) easily checked by cubing, but not so easily found. Still no luck with sqrt(binomial) = hexanomial.

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[math-fun] Re: Betwixt and between
by Richard Guy 05 Jul '04

05 Jul '04

Further to my recent message. I currently have a student who came 9th in the Putnam at age 16. He'a all of 17 now. We hope to write a few joint papers with 70-year age difference betwixt the authors. R. On Sat, 3 Jul 2004, Henry Gould wrote: > > You got ne beat! . . . I will be 76 years young on > August 26. I am still teaching > full time at West Virginia University and am about > to start my 47-th year as > a professor of mathematics here. I know there are > some SeqFans older than you, > but the sequence of SeqFans ages is only a finite > sequence because n+1 > terminates for sure when n is greater than (or > greater than or equal to?) 120. > I cannot prove that 120 is the best upper bound for > this sequence. Is any > SeqFan regular near 90? near 100? I have a friend > who is 99 and surfs the web, > but knows nothing about sequences. > WHO is the youngest SeqFan????? Greater than or > equal to 16?Speak now > or forever hold your peace until your n approaches > 120. > > H (things taken) W (at a time) Gould

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