January 2012 - mathfun@mailman.xmission.com

Re: [math-fun] pizza packing
by Bill Gosper 05 Jan '12

05 Jan '12

Are the exact polynomials available? --rwg I love that m=n sequence 2, 2, 1.942, 1.997, 1.951, 2, 1.942,... State lasciando il settore italiano --------------- on the off-chance that some of you like recreational packing problems, this month's math magic problem is about efficient packings of pizza slices: http://www2.stetson.edu/~efriedma/mathmagic/0112.html erich

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Re: [math-fun] Unique factorization
by Dan Asimov 05 Jan '12

05 Jan '12

These rings are defined as the algebraic integers O_N of the field Q(sqrt(-N)). But in fact O_3 = Z[(sqrt(-3)+1)/2] and O_7 = Z[(sqrt(-7)+1)/2]. In Z[sqrt(-3)] we have (1+sqrt(-3))(1-sqrt(-3) = 2*2, and in Z[sqrt(-7)] we have (1+sqrt(-7))(1-sqrt(-7) = 2*2*2, showing non-unique factorization. Also, I believe Gauss already knew that these 9 rings had unique factorization, and conjectured that they were the only ones; Heegner and Stark each proved this conjecture. --Dan << K.Heegner 1952 & H.Stark 1967: If sqrt(-N) is adjoined to the integers, N>0, then we get a ring enjoying "unique factorization into primes" exactly in the following cases: N = 1,2,3,7,11,19,43,67,163. >> ________________________________________________________________________________________ It goes without saying that .

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[math-fun] pizza packing
by Erich Friedman 05 Jan '12

05 Jan '12

on the off-chance that some of you like recreational packing problems, this month's math magic problem is about efficient packings of pizza slices: http://www2.stetson.edu/~efriedma/mathmagic/0112.html erich

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[math-fun] Fracfac expansions (fractions in factorial base)
by Dan Asimov 04 Jan '12

04 Jan '12

Since high school I've thought factorial base is an interesting way to express real numbers in the unit interval (0,1). That is, since 1/2! + 2/3! + 3/4! + ... = (d/dx((e^x-1)/x))(1) = 1, every x in (0,1) has a unique representation of the form x = Sum_{n=1..oo} c_n/(n+1)! if the integers c_n satisfy 0 <= c_n <= n. E.g., e = 2 + (1,1,1,...) and pi = 3 + (0,0,3,1,5,6,5,0,1,4,7,8,0,...). But I've found very little written on the subject. If anyone can direct me to articles on this subject I'd appreciate it. In particular: Is anything known about fracfac expansions of pi ? Of algebraic numbers like quadratic irrationals? Even rational numbers' fracfac expansions, which clearly terminate, are interesting to look at, especially reciprocal primes. Is there a theorem like the one about Khinchin's constant K for continued fractions? (To wit: For almost every real number, the convergents of its continued fraction have the same constant K = 2.685... as the limit of their geometric mean; see < http://en.wikipedia.org/wiki/Khinchin%27s_constant >.) (Actually, it seems clear that for almost every real number, the coefficients c_n of the fracfac expansion must approach the uniform distribution on {0,1,2,...,n}. More precisely, as n -> oo, the distribution of the ratios c_n/n must approach the uniform distribution on (0,1).) ((( Note: It's a pleasant problem, if you haven't seen it, to calculate f(K) := Sum{n=0..oo} 1/(nK)! ))) --Dan ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Sum of reciprocal integer powers = 3/5 ???
by Dan Asimov 04 Jan '12

04 Jan '12

I'm pretty sure Gareth nailed the sum I was after. And I agree with him that the striking fact that (*) Sum_{k,n=2..oo} 1/k^n = 1 cries out for an immediate explanation, even though Gareth's computational proof can hardly be shortened. Meanwhile, I'm also sure I recall recently reading of something related being = 3/5. How about this: What is Sum_{1/L | L = k^n for some k,n >= 2} ? In other words, the same sum as (*) but with all repetitions removed. --Dan << Gareth wrote: << I wrote: A few years ago I read in some math journal that a certain sum of reciprocal integer powers has an unexpectedly simple sum -- I think it was 3/5. But I can't find the article or a reference to this fact. So I don't know just which reciprocal integer powers are being summed (or whether 3/5 is right). >> I don't think this is what you're after -- it's too easy and the sum isn't 3/5 or much like it -- but sum {n>=2,k>=2} 1/n^k = sum {n>=2} sum {k>=2} 1/n^k = sum {n>=2} 1/n(n-1) = sum {n>=2} 1/(n-1) - 1/n = 1 which is kinda cute. (Query: is there a "trivial" proof that regards each 1/n^k term as a probability or something and thereby makes it instantly obvious that the sum is 1?) >> ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Output puzzle
by Dan Asimov 04 Jan '12

04 Jan '12

The 10 lines below comprise a segment of Mathematica output. Puzzle: What do these numbers represent? --Dan {8, 4, 2, 1} {8, 4, 2, 1, 1} {10, 5, 2, 1, 1} {10, 5, 2, 1, 1, 1} {11, 5, 2, 2, 1, 1} {11, 6, 3, 2, 1, 1} {15, 6, 3, 2, 1, 1} {15, 6, 3, 2, 1, 1, 1} {16, 8, 3, 2, 1, 1, 1} {16, 8, 3, 2, 1, 1, 1, 1} ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Sum of reciprocal integer powers = 3/5 ???
by Dan Asimov 04 Jan '12

04 Jan '12

A few years ago I read in some math journal that a certain sum of reciprocal integer powers has an unexpectedly simple sum -- I think it was 3/5. But I can't find the article or a reference to this fact. So I don't know just which reciprocal integer powers are being summed (or whether 3/5 is right). Can anyone give me a pointer to this result? Thanks, Dan ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Sum of reciprocal integer powers = 3/5 ???
by Dan Asimov 03 Jan '12

03 Jan '12

Thanks, but let me clarify that a) the sum in question is infinite, and b) all powers (whose reciprocals appear in the sum) are >= 2. --Dan Bob Baillie wrote: << i don't know about a sum that equals 3/5, but there are the following sums of harmonic numbers that unexpectedly sum to 2 and 6: http://www.math.ubc.ca/~gerg/papers/downloads/recsum2.pdf http://www.math.ubc.ca/~gerg/papers/downloads/recsum6.pdf bob baillie --- I wrote: << A few years ago I read in some math journal that a certain sum of reciprocal integer powers has an unexpectedly simple sum -- I think it was 3/5. But I can't find the article or a reference to this fact. So I don't know just which reciprocal integer powers are being summed (or whether 3/5 is right). Can anyone give me a pointer to this result? >> >> ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] math-fun Digest, Vol 107, Issue 4
by Tom Duff 03 Jan '12

03 Jan '12

> Strangely, the number 237 has its partisans, some of whom contribute to > this list. See HAKMEM for examples. Personal note: 237 Erskine Avenue in > Toronto was the house I lived in as a young (pre-HAKMEM!) child. Oops, 239, as pointed out by others... (By way of explanation (?), at some point in my childhood we moved next door to 239 Erskine Avenue, after which my memory inexplicably developed a lacuna...) -- Tom Duff. I put bomb in squirrel's briefcase and who gets blown up? Me!

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Re: [math-fun] math-fun Digest, Vol 106, Issue 44
by Henry Baker 03 Jan '12

03 Jan '12

239 At 10:54 AM 1/3/2012, Mike Stay wrote: >I searched google for [inurl:hakmem 237] and got nothing of interest. >More specific pointer? > >On Tue, Jan 3, 2012 at 10:48 AM, Tom Duff <td(a)pixar.com> wrote: >>> Then, by chance I got a couple emails in a row from members of the Cult of >>> 137 [3], and I decided to prove to myself (and to them) that it is easy to >>> make all sorts of meaningless formulas for any number you choose. >> >> Strangely, the number 237 has its partisans, some of whom contribute to this >> list. See HAKMEM for examples. Personal note: 237 Erskine Avenue in Toronto >> was the house I lived in as a young (pre-HAKMEM!) child. >> >> -- >> Tom Duff. Employee number 70068288. >-- >Mike Stay - metaweta(a)gmail.com >http://www.cs.auckland.ac.nz/~mike >http://reperiendi.wordpress.com

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