September 2008 - mathfun@mailman.xmission.com

[math-fun] a(n) = PrimePi(a(n+1))
by Eric Angelini 17 Sep '08

17 Sep '08

Hello Math-Fun, as I get this from SeqFans (which I don't understand) I beg for mercy re-sending my post here. Best, É. --- Certains des destinataires ou tous les destinataires n'ont pas reçu votre message. Objet : a(n) = PrimePi(a(n+1)) Date : 17/09/2008 15:33 Impossible de contacter le(s) destinataire(s) suivant(s) : seqfan(a)ext.jussieu.fr le 17/09/2008 15:32 L'organisation à laquelle le message a été envoyé a indiqué qu'elle ne contenait pas ce compte de messagerie. Vérifiez l'adresse de messagerie du destinataire ou bien contactez le destinataire directement pour lui demander son adresse exacte. < shiva.jussieu.fr #5.1.1 smtp; 550 5.1.1 <seqfan(a)ext.jussieu.fr>... User unknown> --- Hello SeqFans, Could someone please compute a few more (not too big) terms? Best, É. http://www.research.att.com/~njas/sequences/A000720 0,1,2,3,5,11,31,127,... a(n) is the smallest integer equal to PrimePi(a(n+1)) [this is a very, very bad mixture of pidgenglish and maths -- sorry. How could the seq be properly defined?] (done by hand using this table: http://www.research.att.com/~njas/sequences/b000720.txt ) Method (see table below): - start with the first 0 on the right 0 - this 0 comes from 1, on the left - look for the first 1 on the right 1 - this 1 comes from 2, on the left - look for the first 2 on the right 2 - this 2 comes from 3, on the left - look for the first 3 on the right 3 - this 3 comes from 5, on the left - look for the first 5 on the right 5 - this 5 comes from 11, on the left - look for the first 11 on the right 11 - this 11 comes from 31, on the left - look for the first 31 on the right 31 - this 31 comes from 127, on the left - look for the first 127 on the right 127 ... 1 0 <-- 2 1 <-- 3 2 <-- 4 2 5 3 <-- 6 3 7 4 8 4 9 4 10 4 11 5 <-- 12 5 13 6 14 6 15 6 16 6 17 7 18 7 19 8 20 8 21 8 22 8 23 9 24 9 25 9 26 9 27 9 28 9 29 10 30 10 31 11 <-- 32 11 33 11 34 11 35 11 36 11 37 12 38 12 39 12 40 12 41 13 42 13 43 14 44 14 45 14 46 14 47 15 48 15 49 15 50 15 51 15 52 15 53 16 54 16 55 16 56 16 57 16 58 16 59 17 60 17 61 18 62 18 63 18 64 18 65 18 66 18 67 19 68 19 69 19 70 19 71 20 72 20 73 21 74 21 75 21 76 21 77 21 78 21 79 22 80 22 81 22 82 22 83 23 84 23 85 23 86 23 87 23 88 23 89 24 90 24 91 24 92 24 93 24 94 24 95 24 96 24 97 25 98 25 99 25 100 25 101 26 102 26 103 27 104 27 105 27 106 27 107 28 108 28 109 29 110 29 111 29 112 29 113 30 114 30 115 30 116 30 117 30 118 30 119 30 120 30 121 30 122 30 123 30 124 30 125 30 126 30 127 31 <-- 128 31 129 31 130 31 131 32 ... Best, É.

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[math-fun] Mersenne Numbers - correction
by Guy Haworth 17 Sep '08

17 Sep '08

Thanks for the correction, Dan. I should have said that UCLA won the EFF Prize, but not for themselves alone. http://www.youtube.com/watch?v=6dG0Vk2XZMw - 5.40 in, enjoy Previn: You're playing all the wrong notes. Morecambe: I'm playing all the right notes, but not necessarily in the right order! G

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Re: [math-fun] Equation mixing integers and reals
by Dan Asimov 16 Sep '08

16 Sep '08

Arithmetic correction: 2^4 / 4! = 2/3, not 2/15. So my figures (at this point) were off by a factor of 5, no big deal. Sorry about that. --Dan _____________________________________________________________________ "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] Equation mixing integers and reals
by Dan Asimov 16 Sep '08

16 Sep '08

I wrote: << Jim Buddenhagen writes: << This polynomial: x^4 - 48*x^3 - 12*x^2 - 33*x + 1613 has a root very near Pi. How can I know if this is unusual for the size of the coefficients? . . . >> . . . The difference [between Pi and the root} seems astonishingly low, to 6 places: 1.369107 x 10^(-12). But if we consider all monic integer polynomials up to degree 4 using coefficients of absolute value up to 1613, there are . . . 108,441,586,233,841 polynomials to consider, which makes this sound less astonishing. >> Another valid viewpoint might be to consider only monic integer polynomials with sum-of-absolute- coefficients <= 48 + 12 + 33 + 1613 = 1706. For K > 0, the volume V(K) in 4-space of the region where |c_1| + . . . + |c_4| <= K is about the number of lattice points therein. I get V(K) = 2^4 * K^4 / 4! = 2 K^4 / 15, so V(1706) = 1,129,418,361,346 to the nearest integer, or about 1.1 x 10^12, about a trillion. Together with the answer to this question: [QUESTION: What's a good estimate for (maximum root of P) - (minimum root of Q) over all real roots of monic integer polynomials P, Q of degree d whose sum-of-absolute-values- of-coefficients is <= K ???] -- call this difference MM(d;K) -- then we can say the average spacing between such roots is about avspace = MM(4;1706) / 10^12 and we can ask what the expected distance from the nearest root to pi would be; this would be roughly avspace/4. So if MM(4; 1706) is significantly bigger than 4, we can suddenly gasp in surprise over the closeness to pi of the root Jim mentioned. --Dan _____________________________________________________________________ "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] Mersenne 43,112,609 and 37,156,667 are prime
by rcs＠xmission.com 16 Sep '08

16 Sep '08

Bob Baillie noticed this morning that the two recently discovered Mersenne primes have been "announced", after completing several independent checks. The exponents are 43,112,609 and 37,156,667. The larger number was found a few days earlier. The announcement is at http://www.mersenne.org/ and a press release at http://www.mersenne.org/m45and46.htm It's been two years since the last score; this double hit is a nice reward. It would be interesting to look into using the new high-end graphics cards as compute engines for GIMPS. The floating point is only 32-bit, but one card can do multiply-add steps at 150G/second. Rich

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Re: [math-fun] How do I reverse-project 2D points into 3D? (special case)
by Ray Tayek 16 Sep '08

16 Sep '08

At 12:56 PM 9/16/2008, you wrote: >I have 4 2D points in (computer) screen-space, and I need to >reverse-project them back into 3D space. ... http://library.wolfram.com/infocenter/Articles/2794/ http://campar.in.tum.de/Students/SepPoseEstimation http://opencvlibrary.sourceforge.net/Posit will sorta work, but it can diverge or loop. thanks --- vice-chair http://ocjug.org/

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Re: [math-fun] Equation mixing integers and reals
by Eugene Salamin 16 Sep '08

16 Sep '08

----- Original Message ---- From: Simon Plouffe <simon.plouffe(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Tuesday, September 16, 2008 10:54:41 AM Subject: Re: [math-fun] Equation mixing integers and reals Hello, Concerning the general problem of finding an integer relation. Here is an algorithm for 3 values, that is : if we are looking for aX + bY + Z ? 0 where a,b are 2 real numbers and X,Y,Z three integers. do {| a - b |} = c a <- b b <- c od { } is the fractional part. If we iterate this, after a certain amount of iterations we find 3 integers where the equation is near 0. I could generalize this example to more ; Here is the program in Maple : pseudoLLL:=proc(s) local a, b, c, d, nn, k,i, aa, ss, g, h,m,z; aa:=s: nn:=nops(aa): k:=[seq(frac(abs(aa[i]-aa[i+1])),i=1..nn-1),frac(abs(aa[1]-aa[nn]))]; ss:=evalf(k): g:=sort([seq([convert(1+ss[j],string),j],j=1..nn)],lexorder[2]): h:=[seq(g[i][3],i=1..nn)]: m:=[seq(k[h[z]],z=1..nn)]: return(m); end: This will work with a vector of real numbers up to a certain precision. I could make it to work with Maple 11, but they changed the syntax and the calling sequence again for the sort function (#@!$!@#($&!). For n=3 there, it can be worked out this way : L3:=proc(x, y, z) local a, b, c, d, k1, k2, k3, ll; a := x; b := y; c := z; k1 := frac(abs(a - b)); k2 := frac(abs(b - c)); k3 := frac(abs(a - c)); ll := sort([k1, k2, k3]); a := ll[1]; b := ll[2]; c := ll[3]; RETURN(a, b, c) end: Simon Plouffe -------------------------------------------- I don't understand this at all. The return values are real numbers between 0 and 1, not integers. For example, in Maple 12: > L3(evalf(Pi), evalf(exp(1)), evalf(sqrt(2))); 0.304068266, 0.423310826, 0.727379092 Gene

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[math-fun] How do I reverse-project 2D points into 3D? (special case)
by Paul Reiners 16 Sep '08

16 Sep '08

I have 4 2D points in (computer) screen-space, and I need to reverse-project them back into 3D space. I know that each of the 4 points is a corner of a 3D-rotated rigid rectangle, and I know the size of the rectangle. Is it possible to recover the original 3D coordinates from this? If so, how? (I originally heard this problem here: http://stackoverflow.com/questions/76134/how-do-i-reverse-project-2d-points… )

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Re: [math-fun] Equation mixing integers and reals
by Dan Asimov 16 Sep '08

16 Sep '08

Jim Buddenhagen writes: << This polynomial: x^4 - 48*x^3 - 12*x^2 - 33*x + 1613 has a root very near Pi. How can I know if this is unusual for the size of the coefficients? -- I think that it is. I suppose this problem is in the category you describe -- but sorry I don't know how to do it. >> So this of course got me wondering just how close this root is to pi. Mathematica gives this root to 20 places as x = 3.1415926535884241315 . To 20 places, Pi = 3.1415926535897932385 . The difference seems astonishingly low, to 6 places: 1.369107 x 10^(-12). But if we consider all monic integer polynomials up to degree 4 using coefficients of absolute value up to 1613, there are say (3227^4 = 108,441,586,233,841 polynomials to consider, which makes this sound less astonishing. --------------------------------------------------- QUESTION: By the way, is there some nice estimate for the maximum value among all real roots, of all monic integer polynomials of degree <= d, having coefficients of absolute value <= N ??? Call this M(d, N). Any good estimates, or asymptotic ones? --Dan _____________________________________________________________________ "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] Equation mixing integers and reals
by Christian Boyer 16 Sep '08

16 Sep '08

Let f(x , x , ..., x ) = k x + k x + ... + k x 1 2 n 1 1 2 2 n n where k , k , ..., k are n REAL constants 1 2 n and x , x , ..., x are n INTEGERS (positive or negative) 1 2 n If we try to have the best possible approximations of f(x , x , ... x ) = 0 1 2 n do you know good algorithms giving best values of (x , x , ..., x )? 1 2 n When n = 2, a good algorithm is to compute the continued fraction of k / k 2 1 At each step of the algorithm, the continued fraction will give more and more excellent (x , x ). 1 2 But which algorithm(s) for n > 2? Christian.

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