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[math-fun] Pairwise sums/differences are semiprimes.
by Zak Seidov 28 Oct '16

28 Oct '16

Set of integers that all pairwise sums are semiprimes? E.g., s = {1,3,32,54} with pairwise sums 4,33,55, 35,57, 86 all semiprimes Set of integers that all pairwise differences are semiprimes? E.g., s={ 1, 5, 11, 15, 134, 220} with pairwise differences 4,10,14,133,221, 6,10,129,215, 4,123,209 119,205 86 all semiprimes. -- Zak Seidov

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[math-fun] Golomb rulers and 492116
by Tomas Rokicki 27 Oct '16

27 Oct '16

I thought I would share that Gil Dogon and I, inspired by an Al Zimmermann programming contest, went ahead and calculated subquadratic Golomb rulers as reasonably far as we could. We were able to find subquadratic rulers through 492115 marks. At 492116 marks, none of the known (to us) constructions led to a subquadratic ruler. Is it possible that no subquadratic Golomb ruler exists at 492116 marks? Or is it more likely that one exists, and if so, is it something that we will ever explicitly know? -- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --

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[math-fun] Cooper/Zudilin: Hypergeometric modular equations
by Joerg Arndt 26 Oct '16

26 Oct '16

Those of us who appreciate that kind of stuff might find the following entertaining. Shaun Cooper, Wadim Zudilin: Hypergeometric modular equations, http://arxiv.org/abs/1609.07276 Best regards, jj

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[math-fun] Mandelbrot''s (frac-7) snowflake fill
by Bill Gosper 25 Oct '16

25 Oct '16

Howard Cannon used Julian's fractal Fourier expander to make gosper.org/mandelfou.png illustrating that it fills some areas unboundedly faster than others. (dArea/dt is interesting.) The function has no symmetry, i.e., for no (nontrivial) a, b does Z(t) = a(Z(b(t)), and there is no symmetry in the filled areas of winding number 1 of the Fourier approximations, such as illustrated. (Except for the fundamental circle and ellipse. gosper.org/mandelfou-1-2-3.png) Yet for infinitely many discrete samplings, the filled (polygonal) area does have symmetry, e.g, gosper.org/blusno.gif . --rwg

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[math-fun] physicist click bait
by Bill Gosper 23 Oct '16

23 Oct '16

http://www.symmetrymagazine.org/article/july-2015/lhc-physicists-discover-f… trumpets for screenfuls the discovery of two pentaquarks. But a URL at the bottom of the page says Oops, never mind. So why don't they warn readers at the top of the article? --rwg

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[math-fun] 50% gray
by Bill Gosper 22 Oct '16

22 Oct '16

On 2016-10-22 15:07, Eugene Salamin via math-fun wrote: > Let's ask for such a set S as a subset of the unit interval instead of > the entire real line. Then S = S intersect [0,1], and so S must have > measure 1/2. So if S can't be measurable, that completes the proof. > But why is S nonmeasurable? > > -- Gene Julian (quoted without permission): >rwg>I thought I read somewhere that there was no subset of the Reals whose > >intersection with every interval has measure density ½. > >Correct. Lebesgue measure is defined (when using Carathéodory) as the inf of the measures of all countable unions of intervals containing the set. Taking a union of intervals with measure ≤1.01 times the measure of the set, one of the intervals >must have measure ≤1.01 times the measure of its intersection with the set, i.e. in that interval the set has density >99%. rwg>Yet I think I see a nifty one. What gives? >Depends on your proposed construction. Most likely, it either gives a non-measurable set, or gives a set of density 0 or 1 (most recursive constructions do this—almost all points will alternate between being in and out), or is not well-defined. > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Sent: Saturday, October 22, 2016 12:46 PM > Subject: Re: [math-fun] 50% gray > > Are you using the axiom of choice? Or is your set constructible? (Clearly > it's not measurable.) > > Jim Propp I am using hallucinogens in conjunction with the axiom of stupidity. My "solution" was to partition the reals according to the relation of Khinchin's constant to the geometric mean of their partial quotients! (Sorry.) --rwg > > .On Saturday, October 22, 2016, Bill Gosper <billgosper(a)gmail.com> wrote: > >> I thought I read somewhere that there was no subset of the Reals whose >> intersection with every interval has measure density ½. Yet I think I see >> a nifty one. What gives? --rwg >

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[math-fun] 50% gray
by Bill Gosper 22 Oct '16

22 Oct '16

I thought I read somewhere that there was no subset of the Reals whose intersection with every interval has measure density ½. Yet I think I see a nifty one. What gives? --rwg

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[math-fun] https://upload.wikimedia.org/wikipedia/commons/a/ae/Ishihara_19.PNG
by Bill Gosper 22 Oct '16

22 Oct '16

claims to be a red-green deficiency "trick question". You only see the numeral if you're colorblind. But after pasting the image into Mathematica, Rohan and I were unable to find a mapping, e.g., Image[(ImageData@%702) /. {r_, g_, b_, a_} :> {r', g',b',a'}] to reveal the hidden numeral. What's the trick? --rwg

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[math-fun] Number of semiprimes
by Don Reble 22 Oct '16

22 Oct '16

> What is the maximal number of semiprimes among 100 consecutive > numbers? 38? We might never know. Considering only divisibility by small semiprimes, one can find a sequence of only 69 consecutive numbers, 39 of which are not divisible by any semiprime <= 69. For examples, 42117702927300*N + 13684292903, 05, 06, 07, 09, 11, 13, 14, 15, 17, 18, 21, 23, 26, 27, 29, 31, 33, 35, 38, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 57, 59, 61, 62, 63, 66, 67, 69, 71. Maybe for some choice of N, all of these are semiprime. But I despair of finding such an N. The minimum span (last - first + 1) of N semiprimes goes 1,2,3,5,6,7,9,11,13,14,16,17,19,21,... or does it? If that's correct, it (or the incremented sequence) would surely be in the OEIS: where did I go wrong? How does the density of semiprimes vary with the size? -- Don Reble djr(a)nk.ca

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[math-fun] For the mathematician's coffee table: "Mathematics + Art"
by Hilarie Orman 22 Oct '16

22 Oct '16

"Mathematics + Art: A Cultural History" by Lynn Gamwell is beautiful book. The format is large, the printing and the paper are high quality. It is pleasurable to leaf through it (but it is also very heavy --- the cover is slick, I almost dropped it, wear shoes while carrying). My favorite illustration is the last one in the book, before the acknowledgements: the root system of E8. That is available on the Internet, of course, but I wouldn't have thought to look for it had it not been in the book. The illustrations for the modern era are necessarily limited, else the book would go on for several more volumes. I think that the cultural aspect of the book begins to break down in the 20th century, somehow. There is math, there are computers, the whole way of thinking about creativity has shifted. Hilarie

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