April 2008 - mathfun@mailman.xmission.com

[math-fun] Fraction of indivisible numbers
by Steve Witham 04 Apr '08

04 Apr '08

>From: "Schroeppel, Richard" <rschroe(a)sandia.gov> > >product (1 - z^n) for z = 1/2. > >1 - 2^-1 - 2^-2 + 2^-5 + 2^-7 - ... > >The exponents are (negative) pentagonal numbers. >The binary expansion of .2887... should be interesting. Starting after the radix point, 1 zero, 1 one, 2 zeroes, 1 one, 2 zeroes, 4 ones, 1 zero, 3 ones, 6 zeroes, 1 one, 4 zeroes, 8 ones, 1 zero, 5 ones, 10 zeroes, 1 one, 6 zeroes, 12 ones, ... --Steve

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Re: [math-fun] distinct distances
by N. J. A. Sloane 02 Apr '08

02 Apr '08

JZ said: > In particular, can someone explain to me the meaning of the notation > A_1, A_2, D_4, E_8, E_8 + A_1 and so on? Me: These are root lattices and lattices glued together from root lattices - see Conway and Sloane, Sphere Packing, Lattices and Groups (aka SPLAG) > Also there's a lot of reference there to a WD Smith preprint -- does > anyone have a reference to that? Me: I believe this manuscript was in the end withdrawn Neil

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Re: [math-fun] distinct distances
by N. J. A. Sloane 02 Apr '08

02 Apr '08

Joshua, about the Erdos distinct distances problem - did you see my paper on this?  <LI><A NAME="P10161">161. <A>Lattices with Few Distances [<a href="few.ps">postscript</a>, <a href="few.pdf">pdf</a>], [note: some <a href="fewtab.ps"> tables </a> are in a separate file] J. H. Conway and N. J. A. Sloane, J. Number Theory, 39 (1991), pp. 75-90, Also DIMACS Technical Report 90-77, December 1990.</LI> (on my Publication list) Best regards Neil

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Re: [math-fun] Fraction of indivisible numbers
by Fred lunnon 02 Apr '08

02 Apr '08

On 3/30/08, Steve Witham <sw(a)tiac.net> wrote: > oo > ------ > | | 1 > | | 1 - ----- > | | f(i) > i = 1 > > > f(i) = i + 1 > (1/2) (2/3) (3/4) (4/5)... = 0 > > f(i) = (i+1)^2 > (1/4) (8/9) (15/16) (24/25) ... = 0 > > Is the result always zero when f(i) is a positive polynomial of i? > The last equation should apparently read (3/4) (8/9) (15/16) (24/25) ... = 1/2 So no, the product is not always zero --- in fact, taking logarithms and approximating the log by its Taylor expansion log(1-x) = -x + ... , the product is actually nonzero for degree > 1 --- unless one of the factors vanishes "accidentally" because f(i) - 1 happens to have a positive integer root. > f(i) = i'th prime > (1/2) (2/3) (4/5) (6/7) (10/11) (12/13) ... = 0 > > (The fraction of counting numbers that aren't divisible by any prime.) > Does that equation have a name? > > f(i) = 2^i > (1/2) (3/4) (7/8) (15/16) ... ~= 0.288788095087 > This number turns up as the probability that a large random binary matrix is nonsingular. I don't know of a closed form for it --- does anybody else? > Is the result always > 0 when f(i) = k^i and k > 1? > Yes, by the same reasoning as before. > If so, does this put the primes between polynomials and exponentials? > No. A classic result of number theory says that the i-th prime equals i/log(i) (very roughly) --- this is the "Prime Number Theorem", see Hardy & Wright "Introduction to the Theory of Numbers" for a detailed account [and no doubt Wikipedia for a quick 'n' dirty summary]. Fred Lunnon

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[math-fun] New (to me) proof of divergence of the harmonic series
by Joshua Zucker 01 Apr '08

01 Apr '08

Up until now, I only knew a couple of standard proofs that the harmonic series diverges: 1) 1 + 1/2 + (1/3 + 1/4) + (1/5 + ... + 1/8) + ... > 1/2 + 1/2 + 1/2 + 1/2 + ... 2) integral(1/x) = ln x I think that's about it ... But then I saw this very very cool proof idea in what was otherwise a mostly weak book of philosophy of mathematics ( http://www.amazon.com/How-Mathematicians-Think-Contradiction-Mathematics/dp… if you are interested ) and I think it could be said something like this: Suppose it converges. Then let H = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... so 1/2 H = 1/2 + 1/4 + 1/6 + 1/8 + ... and, subtracting, H - 1/2 H = 1 + 1/3 + 1/5 + ... Finally comparing term-for-term, it's clear that 1/2 H > 1/2 H, contradiction, QED. Does anyone have an early citation for this proof, or know who might have thought it up? Byers doesn't give any citation for the idea in his book. Are there any major holes in the proof that need to be filled, or is this basically a good outline? (Since we assume convergence to get the contradiction, I think the manipulations in the middle are OK, right?) Thanks, --Joshua Zucker

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[math-fun] a curious formula
by Simon Plouffe 01 Apr '08

01 Apr '08

Hello, you know probably about this formula : Exp[Pi]-Pi = 19.9990999...? Here is one interesting consequence : Pi = 3 + 1/11- 11^2/34- 23^2/172-... Exp[Pi]= 23 + 1/11- 11^2/34- 23^2/186-... Pi = 3 + 1/11- 11/3- 2/12- 11/6-... Exp[Pi]= 23 + 1/11- 11/3- 2/12- 11/24-... this is one of the gen. continued fraction expansions of these 2 numbers. Simon

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