August 2017 - mathfun@mailman.xmission.com

[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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Re: [math-fun] 50,847,478
by Henry Baker 12 Sep '17

12 Sep '17

= 2*11*2311249. ??? At 03:53 PM 8/29/2017, Tomas Rokicki wrote: >-tom > >-- >-- http://cube20.org/ -- http://golly.sf.net/ --

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[math-fun] Big Splat theory
by James Propp 04 Sep '17

04 Sep '17

Has anyone ever proposed in the Annals of Improbable Research or a similar venue that, in addition to undergoing periodic reversals of its magnetic field, Earth is also subject to occasional short-lasting reversals of its gravitational field? Such a phenomenon would handily explain selective extinction of megafauna: a mouse could survive an upward fall for five seconds followed by a downward fall for five seconds followed by an extreme deceleration, but a mastodon would not fare as well. :-) Jim Propp

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[math-fun] Babylonian trig
by Henry Baker 30 Aug '17

30 Aug '17

I'm still trying to understand what all the fuss is about. Apparently, the Babylonians could do trig w/o angles. I get it: angles (implicitly) require complex exponentials. Perhaps they knew about Gaussian integers, but probably not. But w/o angles, you still need some graded rational approximations to slope; perhaps continued fractions will do? So here's my question (RWG knows this already): Given a continued fraction stream of approximations to a/c, compute the continued fraction stream of approximations to (b/c)=sqrt(1-(a/c)^2).

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[math-fun] Ancient trig
by Brent Meeker 30 Aug '17

30 Aug '17

Scientists recently decoded a clay tablet from ancient Babylonia that dates to around 3,700 years ago, and found that it contains the oldest trigonometric table in the world. https://www.livescience.com/60227-babylonian-clay-tablet-trigonometry.html Brent

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Re: [math-fun] Big Splat theory
by Henry Baker 29 Aug '17

29 Aug '17

So *that's* how pancakes were invented! At 02:20 PM 8/29/2017, James Propp wrote: >I have now irrevocably peed this idea into the global meme-pool: > >https://www.youtube.com/watch?v=JeAnr-EJ48M > >Jim Propp > >On Sat, Jan 11, 2014 at 10:37 PM, James Propp <jamespropp(a)gmail.com> wrote: > >> Has anyone ever proposed in the Annals of Improbable Research or a >> similar venue that, in addition to undergoing periodic reversals of its >> magnetic field, Earth is also subject to occasional short-lasting reversals >> of its gravitational field? >> >> Such a phenomenon would handily explain selective extinction of megafauna: >> a mouse could survive an upward fall for five seconds followed by a >> downward fall for five seconds followed by an extreme deceleration, but a >> mastodon would not fare as well. :-) >> >> Jim Propp

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[math-fun] 50,847,478
by Tomas Rokicki 29 Aug '17

29 Aug '17

-tom -- -- http://cube20.org/ -- http://golly.sf.net/ --

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[math-fun] Fwd: Factoring challenge
by Bill Gosper 28 Aug '17

28 Aug '17

Since braindead xmission.com will doubtless unmask Rohan's shaded hints, I'll just StringReverse them. ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Sun, Aug 27, 2017 at 1:51 PM Subject: Fwd: Factoring challenge To: ..., mathfuneavesdroppers(a)googlegroups.com, ... Rohan occasionally composes cryptically worded puzzles. The hard part is usually to find a sensible interpretation of his problem statement (and hints). He just sent me this atypically satisfying one. I'll add a hint of my own: If you see Homer Simpson, you're on the right track! ---------- Forwarded message ---------- From: Rohan Ridenour <catisfluffy(a)hladkyschool.org> Date: Sat, Aug 26, 2017 at 8:16 PM Subject: Factoring challenge To: Bill Gosper <billgosper(a)gmail.com> x^4+2x^2y^2+y^4-5x^2 Hint: ".scitardauq owt otni srotcaf tI" Doublehint: .ti gnittolp yrT Triplehint: .uoy fo retteb detcepxe I [Just unmasked, pleased to say!-] Fun fact: ".y,x htiw 2^y,2^x ecalper uoy fi rotcaf t'nseod tI" ________ --rwg

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[math-fun] Transfinite ordinals in Shakespeare
by Bill Gosper 28 Aug '17

28 Aug '17

On 2017-08-23 08:50, Hans Havermann wrote: > Check out the Wikipedia entry here: > > https://en.wikipedia.org/wiki/In_saecula_saeculorum > > One gets the sense that any particular translation may or may not > encapsulate what was being translated. The final paragraph touches on > Gareth's "le'alam ul'alme 'almaya". > > >> On Aug 23, 2017, at 10:10 AM, Adam P. Goucher <apgoucher(a)gmx.com> wrote: >> >> I thought I might have been able to get omega^2 with 'in saeculo saeculorum', but unfortunately that merely means 'ages of ages' rather than 'eternity of eternities'. > Way out of my bailiwick here, but wouldn't 'in saeculo saeculorum' be "in (a) century of centuries"? And "ages of ages" be 'saeculae saecularum'? And `in saecula saeculorum' be "in centuries of centuries"? --rwg

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[math-fun] "quadratic residue" division ??
by Henry Baker 26 Aug '17

26 Aug '17

How about this weird form of division (in maxima): qdivide(m,n):=block([q,r,q1,r1], q:floor(m/n), r:m-q*n, /* r is non-negative */ if is(jacobi(r,n)=-1) then (q1:q+1,r1:r-n) else (q1:q,r1:r), [q1,r1]); e.g., (%i25) qdivide(1,19); (%o25) [0, 1] (%i26) qdivide(2,19); (%o26) [1, - 17] (%i27) qdivide(3,19); (%o27) [1, - 16] (%i28) qdivide(4,19); (%o28) [0, 4] (%i29) qdivide(5,19); (%o29) [0, 5] (%i30) qdivide(6,19); (%o30) [0, 6] (%i31) qdivide(7,19); (%o31) [0, 7] (%i32) qdivide(8,19); (%o32) [1, - 11] (%i33) qdivide(9,19); (%o33) [0, 9] (%i34) qdivide(10,19); (%o34) [1, - 9] (%i35) qdivide(11,19); (%o35) [0, 11] (%i36) qdivide(12,19); (%o36) [1, - 7] (%i37) qdivide(13,19); (%o37) [1, - 6] (%i38) qdivide(14,19); (%o38) [1, - 5] (%i39) qdivide(15,19); (%o39) [1, - 4] (%i40) qdivide(16,19); (%o40) [0, 16] (%i41) qdivide(17,19); (%o41) [0, 17] (%i42) qdivide(18,19); (%o42) [1, - 1] (%i43) qdivide(19,19); (%o43) [1, 0] If we utilize this form of division for base conversion to a prime base p=3(mod 4), we automatically get a representation whose digits are +-r, where r is a quadratic residue (mod p). One obvious property: perfect +- symmetry: negation means negating every digit. Questions: Does anything interesting happen when this form of division is used in other places -- e.g., gcd algorithms? Does this type of division induce a slightly different "norm" s.t. the norm of the remainder is strictly < the norm of the divisor? I suspect that there is a close relationship with division in the Gaussian integers, but I haven't checked this yet.

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