August 2009 - mathfun@mailman.xmission.com

[math-fun] factoring almost completely monic polynomials
by Erich Friedman 08 Aug '09

08 Aug '09

for integers m and n, i've been factoring (over the integers) the polynomials: P = (x^n-1) / (x-1) - m obviously when m = (k^n-1)/(k-1) for some integer k, there is a linear factor. i'm interested in the cases when P factors without a linear factor. i've only found 6 cases: (x^8-1)/(x-1) - 3 = (x^3 + x – 1)(x^4 + x^3 + x + 2) (x^14-1)/(x-1) - 4 = (x^3 + x^2 – 1)(x^10 + x^8 + x^7 + 2x^5 + x3 + 2x^2 – x + 3) (x^5-1)/(x-1) + 11 = (x^2 + 3x + 4)(x^2 – 2x + 3) (x^9-1)/(x-1) + 19 = (x^4 – x^3 + x^2 – 3x + 4)(x^4 + 2x^3 + 2x^2 + 4x + 5) (x^11-1)/(x-1) - 23 = (x^2 + x + 2)(x^8 – x^6 + 2x^5 + x^4 – 4x^3 + 3x^2 + 6x – 11) (x^6-1)/(x-1) - 56 = (x^2 – x + 5)(x^3 + 2x^3 – 2x – 11) is there some rhyme or reason for these, or are they just random? erich

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[math-fun] at exactly 12h34
by Simon Plouffe 07 Aug '09

07 Aug '09

Hello (this is a serious posting), at exactly 12h34 and 56 seconds, it will be 1 2 3 4 5 6 7 8 9. this is the french format : 7 août 2009, : 07-08-09. don't miss this historic moment! Simon Plouffe

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Re: [math-fun] pebbles via computer
by Mark Niemiec 06 Aug '09

06 Aug '09

James Propp<jpropp(a)cs.uml.edu> wrote: > If Conway were still posting to math-fun, he'd probably reply > that I'd be better off just making numbered counters and moving > them around. His early work on the Game of Life was done by > hand, and he's of the opinion that some of the insights he > gained from this couldn't have been gained if he'd just had > a computer do the simulations. Most of my early discoveries in Life were also done by moving counters around on a board. And once personal computers became readily available, most of my subsequent ones have been done by single-stepping patterns to observe the state transitions of single cells (rather than just running things at high speed for hundreds or thousands of generations). Sometimes, it can be useful to try to smash a glider into a pattern in every possible orientation (which is amenable to massive parallelization and iteration techiques), but much more often, problems require great precision. For example, synthesis of an object from gliders often requires a specific cell to be turned on or off at a specific time. This requires careful analysis of the entire history of specific cell and its immediate environment, and often requires complicated Rube-Goldberg style machinations. This requires finesse rather than brute force. David Buckingham, who has probably done most of the work on synthesizing most small objects (both simple and complicated) from gliders, would likely express similar sentiments. Many of his syntheses require making slight adjustments to random mushes that occur dozens or even hundreds of generations after gliders collide - which necessarily require observing such interactions in detail one generation at a time, something easy to do when advancing patterns slowly by hand (using counters or a computer), but which would be totally missed when running patterns at high speed. -- Mark D. Niemiec <mniemiec(a)gmail.com>

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Re: [math-fun] L versus 2L
by rwg＠sdf.lonestar.org 05 Aug '09

05 Aug '09

DanA> Interesting that "pace" means 2 steps. > > It certainly also means just 1 step, in common parlance. > > Just one of those little contradictions of language. > Here is def. 3 for pace, according to < www.m-w.com >: > > 3 a : step 2a(1) b : any of various units of distance based on the length of a human step > > and this is def. 2a(1) for step: > > 2 a (1) : an advance or movement made by raising the foot and bringing it down elsewhere > > --Dan > The off-line (and far superior) MW gives 3 a : a movement of the foot over a space to a new position in walking, running, or dancing : STEP *took a pace or two in the room Guy McCrone* b (1) : the space traversed by one step used as an indefinite unit of measure *cannot go five paces without seeing some wretched object Irish Digest* (2) : any of various units of distance based on the length of a human step at a specified time (as for quick time 30 inches and for double time 36 inches) see ROMAN PACE roman pace: an ancient Roman unit of length that is equal to five Roman feet or 4.85 English feet and is measured in pacing from the heel of one foot to the heel of the same foot when it next touches the ground >From the MW etymology, apparently passuum is genitive. We were taught that "mile" came from millia passus, which gets lots of Google hits in Latin texts, but MW likes milia. --rwg ROMAN PACE CAMPANERO

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Re: [math-fun] factoring almost completely monic polynomials
by Dan Asimov 03 Aug '09

03 Aug '09

What are the cases you know that don't work? --Dan Erich wrote: << for integers m and n, i've been factoring (over the integers) the polynomials: P = (x^n-1) / (x-1) - m obviously when m = (k^n-1)/(k-1) for some integer k, there is a linear factor. i'm interested in the cases when P factors without a linear factor. i've only found 6 cases: [details re 5, 6, 8, 9, 11, 14 omitted] is there some rhyme or reason for these, or are they just random? >> _____________________________________________________________________ "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] L versus 2L
by Dan Asimov 02 Aug '09

02 Aug '09

Is this the "Webster's Third New International" ? In an interesting etymological note, RWG wrote: << . . . The off-line (and far superior) MW gives . . . >>> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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