September 2017 - mathfun@mailman.xmission.com

Re: [math-fun] Prime factors of 10^N + 1
by Dan Asimov 26 Sep '17

26 Sep '17

One evident pattern is that after 10^1 + 1 = 11 and 10^2 + 1 = 101, there are no further primes (at least up through 10^200 + 1). Can it be true that for every positive integer b >= 2, there is a integer N(b) > 0 such that n >= N(b) => b^n + 1 is not prime ??? Or is this not plausible? If it *is* plausible, is it subject to vast generalization (plausibly)? —Dan Marc LeBrun wrote: ----- > On Sep 25, 2017, at 5:30 PM, James Davis <lorentztrans(a)gmail.com> wrote: > > Thanks! That saves me a week. Then perhaps I can waste you some more time? So square-free, the exceptions pop out! By eyeball it seems like 11^2 is a divisor when n = an odd multiple of 11. And 7^2 divides n=21 and 63, and there's even a 13^2 for n=39 (no doubt all these are related to n=3 being 7 11 13?) What is the smallest n divisible by a cube? By two distinct squares? By any square that's relatively prime to 1001? -----

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Re: [math-fun] Prime factors of 10^N + 1
by Dan Asimov 26 Sep '17

26 Sep '17

The AMS makes it rather difficult for an individual to learn how to get ahold of this e-book! —Dan ----- >From https://homes.cerias.purdue.edu/~ssw/cun/index.html The Cunningham Project seeks to factor the numbers b^n +- 1 for b = 2, 3, 5, 6, 7, 10, 11, 12, up to high powers n. The Cunningham tables are the tables in the book *"Factorizations of b^n +- 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers,*" by John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff, Jr., American Mathematical Society, Providence, Rhode Island, third edition, 2002. Although the first two editions were published as paper books, the third edition is an electronic book available at http://www.ams.org/books/conm/022/ . -----

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[math-fun] Prime factors of 10^N + 1
by Dan Asimov 26 Sep '17

26 Sep '17

Decided to list these just for the heck of it. Some amusing almost-patterns for 10^0 + 1 through 10^19 + 1: 1 = 1 11 = 11 101 = 101 1001 = 7 * 11 * 13 10001 = 73 * 137 100001 = 11 * 9091 1000001 = 101 * 9901 10000001 = 11 * 909091 100000001 = 17 * 5882353 1000000001 = 7 * 11 * 13 * 19 * 52579 10000000001 = 101 * 3541 * 27961 100000000001 = 11 * 11 * 23 * 4093 * 8779 1000000000001 = 73 * 137 * 99990001 10000000000001 = 11 * 859 * 1058313049 100000000000001 = 29 * 101 * 281 * 121499449 1000000000000001 = 7 * 11 * 13 * 211 * 241 * 2161 * 9091 10000000000000001 = 353 * 449 * 641 * 1409 * 69857 100000000000000001 = 11 * 103 * 4013 * 21993833369 1000000000000000001 = 101*9901*999999000001 10000000000000000001 = 11*909090909090909091 10^20 + 1 is too large for my software. —Dan

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[math-fun] I just found this and I should sleep on it before posting,
by Bill Gosper 25 Sep '17

25 Sep '17

but just in case it's still mysterious in the "morning": What is this? QPolyGamma[0, 1/2, q] + QPolyGamma[0, 1/2 - (I π)/Log[q], q] == -Log[(1 - q)/(1 + q)] + QPolyGamma[0, 1/2, q^2] q-Euler's constants? In[1199]:= PolyGamma[0, 1/2] // FunctionExpand Out[1199]= -EulerGamma - Log[4] Empirically true in the 1st quadrant of the unit disk, and in the whole 4th quadrant. Does it generalize? Does q-digamma have a Jacobi imaginary transformation? zzzzzzzzzz.... --rwg

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[math-fun] WolframAlpha app
by James Propp 25 Sep '17

25 Sep '17

Mathematica-using iPhone users: Have any you tried the WolframAlpha app? Does it let you run real Mathematica code? I should probably just spend the $2.99 and find out for myself what the app can do, but I thought I'd ask first. Jim Propp

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[math-fun] Tetris runs in Conway's Life !
by Henry Baker 24 Sep '17

24 Sep '17

FYI -- I'm not certain how the user interacts in order to "play" the Tetris game, tho... https://developers.slashdot.org/story/17/09/23/0611201/tetris-recreated-in-… 'Tetris' Recreated In Conway's 'Game of Life' Posted by EditorDavid on Saturday September 23, 2017 @07:29PM from the tinkering-with-tetrominoes dept. In 1970 mathematician John Conway created rules for the "Game of Life," a now famous "zero-player game" where a grid of cells evolves (following Conway's rules) from an initial state proposed by the player. In 2013 someone challenged readers of StackExchange's "Programming Puzzles & Code Golf" section to devise an initial state "that will allow for the playing of a game of Tetris." https://codegolf.stackexchange.com/questions/11880/build-a-working-game-of-… An anonymous Slashdot reader reports that "This challenge sat around, gathering upvotes but no answer, for four years. Then, it was answered." Citing the work of seven contributors, a massive six-part response says their solution took one and a half years to create, and "began as a quest but ended as an odyssey." The team created their own assembly language, known as QFTASM (Quest for Tetris Assembly) for use within Conway's mathematical universe, and then also designed their own processor architecture, and eventually even a higher-level language that they named COGOL. Their StackExchange response includes a link to all of their code on GitHub, as well as to a page where you can run the code online. https://codegolf.stackexchange.com/revisions/142673/14 https://codegolf.stackexchange.com/a/142676/2867 https://github.com/QuestForTetris http://play.starmaninnovations.com/qftasm/#jllHdnBGSP One StackExchange reader hailed the achievement as "the single greatest thing I've ever scrolled through while understanding very little."

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Re: [math-fun] Fwd: color illusion
by Bill Gosper 22 Sep '17

22 Sep '17

On 2017-09-22 07:59, Fred Lunnon wrote: > << There is no local cue distinguishing inside from outside of the big loop! >> > > It ain't so: the magenta on the exterior side is broken by cyan blobs which > nearly touch the black ribbon, whereas on the interior they are more distant. > > WFL Nice catch! That also explains why the yellow stars look red on the -outside-. I thought it was because they were yellow! --rwg > > > > On 9/22/17, Bill Gosper <billgosper(a)gmail.com> wrote: >> Out of exasperation with xmission.com's proscription of attachments, I sent >> the following just to the Eavesdroppers. >> >> gosper.org/redliner.nb contains two images of a TrinskyXYPlot of a slightly >> squashed Penrose marking. >> Here's just the png. <http://gosper.org/redliner.png> >> Expand the .nb window until the images stop expanding. Note the big >> blackish "star" seems to have a bright red inner lining. >> Stars of other colors seem to have reddish(?) inner or outer linings which >> disappear on close inspection. >> The bright red is illusory as well. >> >> But wait a minute! There is no local cue distinguishing inside from >> outside of the big >> loop! Why do we see it inside of both the deep concavities and the >> prominent convexities?? >> The loop is cropped without even closing! >> >> Without due credit, I used Jack Holloway's superior rendition of this >> pattern on my G4G12 Exchange Gift card <http://gosper.org/Bill Gosper >> Card.pdf>. >> I'm bugging everybody again, having just noticed two unforgivable typos: >> Ziegler Hunts is misspelled, and worse, the URL to the Minskys and Trinskys >> pdf >> (+ ton of images), which should be gosper.org/Minskys/ . --rwg >> _______________________________________________

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[math-fun] Fwd: color illusion
by Bill Gosper 22 Sep '17

22 Sep '17

Out of exasperation with xmission.com's proscription of attachments, I sent the following just to the Eavesdroppers. gosper.org/redliner.nb contains two images of a TrinskyXYPlot of a slightly squashed Penrose marking. Here's just the png. <http://gosper.org/redliner.png> Expand the .nb window until the images stop expanding. Note the big blackish "star" seems to have a bright red inner lining. Stars of other colors seem to have reddish(?) inner or outer linings which disappear on close inspection. The bright red is illusory as well. But wait a minute! There is no local cue distinguishing inside from outside of the big loop! Why do we see it inside of both the deep concavities and the prominent convexities?? The loop is cropped without even closing! Without due credit, I used Jack Holloway's superior rendition of this pattern on my G4G12 Exchange Gift card <http://gosper.org/Bill Gosper Card.pdf>. I'm bugging everybody again, having just noticed two unforgivable typos: Ziegler Hunts is misspelled, and worse, the URL to the Minskys and Trinskys pdf (+ ton of images), which should be gosper.org/Minskys/ . --rwg

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Re: [math-fun] Thue Morse from 2F1
by Bill Gosper 22 Sep '17

22 Sep '17

Mike Stay wrote On Wed, Sep 20, 2017 at 6:06 PM, Bill Gosper <billgosper(a)gmail.com> wrote: http://mathworld.wolfram.com/Thue-MorseSequence.html gives (if simplified) ThueMorse@n == Mod[3^n*Binomial[1/2, n]*Hypergeometric2F1[3/2, -n, 3/2 - n, -1/3]/2 - 1/2,2] HtF? --rwg Amazing: In[599]:= Series[(1+Sqrt[(1-3x)/(1+x)])/(2(1+x)),{x,0,33}] Out[599]= 1-2 x+2 x^2-3 x^3+2 x^4-5 x^5-x^6-14 x^7-22 x^8-69 x^9-163 x^10-440 x^11-1145 x^12-3068 x^13-8230 x^14-22307 x^15-60790 x^16-166685 x^17-459307 x^18-1271480 x^19-3534115 x^20-9859574 x^21-27598756 x^22-77490473 x^23-218183521 x^24-615902900 x^25-1742738476 x^26-4942022649 x^27-14043034702 x^28-39979680749 x^29-114020882009 x^30-325721340062 x^31-931921909078 x^32-2670196518173 x^33+O[x]^34 In[600]:= Mod[CoefficientList[%,x],2] Out[600]= {1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0,1} I notice that SeriesCoefficient boggles here instead of finding the 2F1. I wonder how SW got it. --rwg -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Thue Morse from 2F1
by Bill Gosper 21 Sep '17

21 Sep '17

http://mathworld.wolfram.com/Thue-MorseSequence.html gives (if simplified) ThueMorse@n == Mod[3^n*Binomial[1/2, n]*Hypergeometric2F1[3/2, -n, 3/2 - n, -1/3]/2 - 1/2,2] HtF? --rwg

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