October 2015 - mathfun@mailman.xmission.com

[math-fun] Fwd: Re: Seven-Planet Somsky Gears
by rwg 26 Oct '15

26 Oct '15

A 10 yr old friend solves the mystery (and makes me feel stupid). --rwg -------- Original Message -------- SUBJECT: Re: [math-fun] Seven-Planet Somsky Gears DATE: 2015-10-26 10:11 TO: rwg <rwg(a)sdf.org> Don't think so. According to the Shapeways page, the red gear is split into two parts. -Me! On Tue, Oct 20, 2015 at 12:32 PM, rwg <rwg(a)sdf.org> wrote: > (Excepting the crank), was it 3D-printed pre-assembled?? > How did he put it together? --rwg > > On 2015-10-19 17:10, William R Somsky wrote: > About all I can say is: COOL! > > Seeing it as a computer generated graphic is one thing, but to see it > in real, physical form? WOW! :-) > > I'm kinda amazed that the three-tooth gear worked! Probably wouldn't > have if you hadn't made them all double-helical. > > Well done! > > - Bill Somsky > > On 2015-10-19 06:39, M. Oskar van Deventer wrote: > Hi Mathfun friends, > > See here a 3D-printed version of the Seven-Planet Somsky Gears. > https://www.youtube.com/watch?v=3iyO-OvIS_k [1] > https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears [2] > > Someone at the Twisty Puzzles Forum asked the question "Is there a link to the arguments for the maximum number of planets?" > http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755 [3] > > Enjoy! > > Oskar We do! Links: ------ [1] https://www.youtube.com/watch?v=3iyO-OvIS_k [2] https://www.shapeways.com/product/SPGDQSRLE/seven-planet-somsky-gears [3] http://twistypuzzles.com/forum/viewtopic.php?p=342755#p342755

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Re: [math-fun] Primes of the form 1+2*p^k
by Henry Baker 26 Oct '15

26 Oct '15

Thanks! p=1 (mod 3) also kills off 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, etc. p^2=1 (mod 3) [p=2=-1 (mod 3)] kills off even powers of the other primes /= 3. Hans Havermann suggests: http://oeis.org/A005234 'Primorial primes' (I'd never heard of this term before) are interesting, but I'm also allowing any subsequence of the primes, so long as they contain 2. At 11:09 AM 10/24/2015, James Buddenhagen wrote: >1+2*7^k is always divisible by 3 > >On Sat, Oct 24, 2015 at 11:55 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Dumb questions re primes. >> >> Primes of the form 1+2^k are quite rare. ;-) >> >> Primes of the form 1+2*3^k seem to be less rare. >> >> Primes of the form 1+2*5^k seem to get rarer. >> >> Primes of the form 1+2*7^k seem to be quite rare. (I don't have a fast >> machine, but I'm having trouble finding even one.) >> >> Primes of the form 1+2*11^k seem to be less rare. >> >> Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are >> odd primes (i.e., primes to the 1st power only) ? >> >> (Perhaps these primes should be called "Euclid primes" after Euclid's >> proof of the infinite # of primes -- if they have no other name?) >> >> Anything known about these distributions? >> >> Also, is the discrete log particularly cheap to compute for any of these >> prime forms?

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Re: [math-fun] HB desires for provable primes
by Henry Baker 26 Oct '15

26 Oct '15

Very interesting! Thanks for the reference, Warren! At 08:30 PM 10/25/2015, Warren D Smith wrote: >sounds like you were seeking this: > >J. Pintz, W. L. Steiger and E. Szemeredi: >Infinite sets of primes with fast primality tests and quick >generation of large primes, Math. Comput. 53,187 (1989) 399-406 > >http://www.ams.org/journals/mcom/1989-53-187/S0025-5718-1989-0968154-X/S002…

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[math-fun] HB desires for provable primes
by Warren D Smith 26 Oct '15

26 Oct '15

sounds like you were seeking this: J. Pintz, W. L. Steiger and E. Szemeredi: Infinite sets of primes with fast primality tests and quick generation of large primes, Math. Comput. 53,187 (1989) 399-406

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Re: [math-fun] There's probably a simple derivation of this,
by Bill Gosper 26 Oct '15

26 Oct '15

The q -> E^(-2 Pi) special case of Sum[DivisorSigma[1,2*n+1] q^n,{n,0,∞}] is way simpler than any of the three eta'/eta log derivatives that comprise it. So I went after the q->E^(-4Pi/3) case: 1 + 2 n Sum[--------------------------------------, {n, 0, ∞}] -> (4 n Pi)/3 -(4/3) (1 + 2 n) Pi E (1 - E ) 1/4 2 (2 Pi)/3 1 4 Sqrt[3] (1 - Sqrt[2] + 3 + Sqrt[3]) E Gamma[-] 4 ----------------------------------------------------------- 3 512 Pi (The quadrinomial optionally factors into binomials. 1 - Sqrt[2] + 3^(1/4) + Sqrt[3] == (1 + 3^(1/4)) (Sqrt[2] + 3^(1/4))/ ((1 + Sqrt[2]) (-1 + 3^(1/4)) (Sqrt[2] + Sqrt[3])). ) But I was not able to derive this sum from the three eta'/eta quoted in the Sum[DivisorSigma[1,2*n+1] q^n,{n,0,∞}] below. I had eta'/eta (i/3) and (2i/3), but to get (4i/3) I had to differentiate Jacobi's Aequatio, which invoked eta'(i/3), eta'(2i3), eta(i/3), eta(2i/3), and eta(4i/3). This created a truly intractable surdstorm. It was not even possible to numerically circumvent RootReduce and reconstruct a simplification with RootApproximant. So I simply RootApproximated the above rhs, from which I reconstructed DedekindEta'[(4 I)/3]==(1/(\[Pi]^(11/4)))((3 I 3^(1/8) (-1+3^(1/4))^2 Sqrt[1+3^(1/4)] \[Pi]^2 Gamma[1/4])/(32 2^(2/3) (-1+3^(1/8))^2 (1+3^(1/8))^2 (Sqrt[2]-3^(1/4))^(3/8) (1+Sqrt[3])^(13/24) (2+Sqrt[2]+Sqrt[3]+Sqrt[6])^(1/4))+(I (-63-6 Sqrt[2]+27 3^(1/4)+18 Sqrt[2] 3^(1/4)-9 Sqrt[3]+21 3^(3/4)+14 Sqrt[2] 3^(3/4)-30 Sqrt[6]) Gamma[1/4]^5)/(1024 2^(2/3) 3^(7/8) (-1+3^(1/8))^2 (1+3^(1/8))^2 (Sqrt[2]-3^(1/4))^(3/8) Sqrt[1+3^(1/4)] (1+Sqrt[3])^(13/24) (2+Sqrt[2]+Sqrt[3]+Sqrt[6])^(1/4))) --rwg Speaking of the Aequatio, Mike Hirschhorn has come up with two nifty derivations and a striking observation: gosper.org/billg102615.pdf For 𝜙 and ψ see the earlier draft gosper.org/billg102115-2.pdf --rwg On Wed, Oct 21, 2015 at 3:15 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > but I sure missed it. > Pi 1 4 > 2 E (-)! > 1 + 2 n 4 > Sum[------------------------------, {n, 0, oo}] == ----------- > 2 n Pi -2 (1 + 2 n) Pi 3 > E (1 - E ) Pi > > I got this as the q -> E^(-2 Pi) special case of > Sum[DivisorSigma[1,2*n+1] q^n,{n,0,∞}]== > Sum[(1 + 2 n) q^n/(1 - q^(1 + 2 n)), {n, 0, ∞}] == ( > I (Derivative[1][DedekindEta][-((I Log[q])/(4 \[Pi]))]/ > DedekindEta[-((I Log[q])/(4 \[Pi]))] - ( > 3 Derivative[1][DedekindEta][-((I Log[q])/(2 \[Pi]))])/ > DedekindEta[-((I Log[q])/(2 \[Pi]))] + ( > 2 Derivative[1][DedekindEta][-((I Log[q])/\[Pi])])/ > DedekindEta[-((I Log[q])/\[Pi])]))/(2 \[Pi] Sqrt[q]) > > As suggested by Sunday's "notational modularity" item, I have a bunch more > special values of eta and eta' to crank into this. (I bet you can hardly > wait.) > Also, eta[q], eta[q^2], and eta[q^4] are polynomially related by Jacobi's > aequatio identica satis abstrusa > <https://www.google.com/search?client=safari&rls=en&q=aequatio+identica+sati…> > , which should have an eta' analog. --rwg > Vaguely related: > ArcLength[{t, Sqrt[Cos[t]]}, {t, -π/2, π/2}] > == > Integrate[Sqrt[1 + (1/4)*Sin[t]*Tan[t]], {t, -Pi/2, Pi/2}] > but the ISC says HUH? Really? (Connection: > Area under that arc = π^(3/2)/(2 Sqrt[2] (1/4)!^2). ) >

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Re: [math-fun] Primes of the form 1+2*p^k
by Henry Baker 25 Oct '15

25 Oct '15

1+2*251^251 is prime. Lisp's integer-length returns 2002 bits. 251 = 1+2*5^3 and is prime. Kinda cool!

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Re: [math-fun] Primes of the form 1+2*p^k
by Henry Baker 25 Oct '15

25 Oct '15

My goal was "inverting" Pratt succinct certificates of primeness, by constructing new primes from primes that already have Pratt certificates. Primes of the form 1+2*p^k, where p already has a certificate, can be given a certificate which is an extension of the certificate of p. These primes aren't just "primes with high probability", but actual provable primes. The problem is that finding such extensions gets harder & harder. For example, primes of the form 1+2*17^i are quite rare (i=47 works). Primes of the form 1+2*47^i are also quite rate (i=175 works). This means that trying to force an extension for a particular Pratt prime is going to be quite wasteful; better to keep the search very wide. At 06:49 AM 10/25/2015, James Propp wrote: >1+2*5^0=3 is prime, >1+2*5^3=251 is prime, >1+2*5^251 is prime, >... ? ... > >(Maybe this progression was implicit in Henry's post, but I thought it >deserved explicit mention.) > >Jim Propp > >On Sunday, October 25, 2015, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> 1+2*251^251 is prime. Lisp's integer-length returns 2002 bits. >> >> 251 = 1+2*5^3 and is prime. >> >> Kinda cool!

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Re: [math-fun] H.Baker's primes
by Warren D Smith 25 Oct '15

25 Oct '15

What HB is calling primes of form 1 + 2*p1*p2*p3... if I understand him aright, these are exactly the odd primes P such that (P-1) is squarefree. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] flexatubes
by Warren D Smith 25 Oct '15

25 Oct '15

I think a similar idea was used by the Wright brothers. That had a biplane wing built like a somewhat-flexible box in a manner with some resemblance to flexatube. By performing a sort of helical twist to that box, you controlled the airplane. The Wright brothers did not use flaps on their wings, they used "wing warping" of the "box" shaped biplane wing. Now actually, I think their solution was simpler and maybe more robust than flaps, at least for early biplanes. Even though the Wrights later switched to flaps and the whole wing warping idea is no longer used, one should admire its elegance. And maybe(?) you could argue the idea still is in heavy use -- by birds.

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[math-fun] Primes of the form 1+2*p^k
by Henry Baker 25 Oct '15

25 Oct '15

Dumb questions re primes. Primes of the form 1+2^k are quite rare. ;-) Primes of the form 1+2*3^k seem to be less rare. Primes of the form 1+2*5^k seem to get rarer. Primes of the form 1+2*7^k seem to be quite rare. (I don't have a fast machine, but I'm having trouble finding even one.) Primes of the form 1+2*11^k seem to be less rare. Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ? (Perhaps these primes should be called "Euclid primes" after Euclid's proof of the infinite # of primes -- if they have no other name?) Anything known about these distributions? Also, is the discrete log particularly cheap to compute for any of these prime forms?

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