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[math-fun] Yellows = non-Yellows
by Éric Angelini 18 Nov '19

18 Nov '19

Hello Math-Fun, S is the lexicographically earliest seq of distinct integers [separated only by a comma, and starting with a(1) = 1] such that the yellow digits appear in the exact same order as the non yellow digits. A yellow digit is a digit in contact with at least one comma of S. Question: For which a(n) will the quantity of yellow digits equal the quantity of non yellow digits [if such an a(n) exists]? S starts like this, if I am not wrong: S = 1,2,3,4,5,6,78,9,10,11,12,13,...,97,98,99, 110,120,130,140,150,160,170,180,190,111, 100,112,113,114,121,115,131,116,141,117,... (S is perhaps already in the OEIS -- but I could not find it; anyway the question remains). Best, É. à+ É. Catapulté de mon aPhone

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[math-fun] Symmetric Angle Doubling Formulas for JacobiSN | CN | DN .
by Brad Klee 16 Nov '19

16 Nov '19

Cross-Post from Wolfram Community: https://community.wolfram.com/groups/-/m/t/1826418 News to me. --Brad

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[math-fun] the impressive speed of primecount
by Simon Plouffe 16 Nov '19

16 Nov '19

Hello, there is a little jewel of program called primecount (easy to find on Windows, mac or linux). It counts the primes. Here is a table of results (see below), from 10^1 to 10^20 with primecount and mathematica : mathematica stops giving answers after 10^14 and takes 26.2656 seconds to find PrimePi[10^14] when primecount takes only : 0.069 seconds (!). This is 431 times faster. I do not mention Maple for decency, it is too slow by far, Pari-Gp is not bad but limited in range too. The -g option stands for the Gourdon method (apparently the fastest). You can download the program here : https://github.com/kimwalisch/primecount it comes with a short but very useful explanation. See for yourself : results are astonishing. The recent record result of Pi(10^27) is mainly due to this program. See http://oeis.org/A006880 for more details. Last thing : the program is 1.52 megs in size and takes all the cpus/threads you may have on one machine automatically. primecount -g 10^1 --time; 4 seconds: 0.000 primecount -g 10^2 --time; 25 seconds: 0.000 primecount -g 10^3 --time; 168 seconds: 0.000 primecount -g 10^4 --time; 1229 seconds: 0.000 primecount -g 10^5 --time; 9592 seconds: 0.000 primecount -g 10^6 --time; 78498 seconds: 0.000 primecount -g 10^7 --time; 664579 seconds: 0.000 primecount -g 10^8 --time; 5761455 seconds: 0.001 primecount -g 10^9 --time; 50847534 seconds: 0.001 primecount -g 10^10 --time; 455052511 seconds: 0.003 primecount -g 10^11 --time; 4118054813 seconds: 0.007 primecount -g 10^12 --time; 37607912018 seconds: 0.015 primecount -g 10^13 --time; 346065536839 seconds: 0.026 primecount -g 10^14 --time; 3204941750802 seconds: 0.069 primecount -g 10^15 --time; 29844570422669 seconds: 0.204 primecount -g 10^16 --time; 279238341033925 seconds: 0.647 primecount -g 10^17 --time; 2623557157654233 seconds: 2.276 primecount -g 10^18 --time; 24739954287740860 seconds: 8.334 primecount -g 10^19 --time; 234057667276344607 seconds: 35.579 primecount -g 10^20 --time; 2220819602560918840 seconds: 155.561 Timing[PrimePi[10^1]] {0., 4} Timing[PrimePi[10^2]] {0., 25} Timing[PrimePi[10^3]] {0., 168} Timing[PrimePi[10^4]] {0., 1229} Timing[PrimePi[10^5]] {0., 9592} Timing[PrimePi[10^6]] {0., 78498} Timing[PrimePi[10^7]] {0., 664579} Timing[PrimePi[10^8]] {0., 5761455} Timing[PrimePi[10^9]] {0., 50847534} Timing[PrimePi[10^10]] {0.046875, 455052511} Timing[PrimePi[10^11]] {0.1875, 4118054813} Timing[PrimePi[10^12]] {1.03125, 37607912018} Timing[PrimePi[10^13]] {5.34375, 346065536839} Timing[PrimePi[10^14]] {26.2656, 3204941750802} Have fun !, Simon Plouffe

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[math-fun] Fred Lunnon's Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.
by Neil Sloane 16 Nov '19

16 Nov '19

I was shocked to see that the sequences in his paper, which are A005318 <https://oeis.org/A005318>, A006754 <https://oeis.org/A006754>, A006755 <https://oeis.org/A006755>, A006756 <https://oeis.org/A006756>, A006757 <https://oeis.org/A006757>, A195683 <https://oeis.org/A195683>, A195684 <https://oeis.org/A195684> have not been extended in 33 years - all we have are the terms Fred computed himself in 1986 (except for the first one, A005318, which is the classic Conway-Guy sequence). Maybe someone would like to extend them!

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[math-fun] Excellent Wikipedia hypocycloid animation
by Bill Gosper 10 Nov '19

10 Nov '19

https://upload.wikimedia.org/wikipedia/commons/d/da/Rolling_Hypocycloids.gif This is the decacuspid (n=10) case. What happens when n➝∞? Why do all the curves coincide only at 3 o'clock and 9 o'clock? Why isn't https://en.wikipedia.org/wiki/Epicycloid nearly as complete as https://en.wikipedia.org/wiki/Hypocycloid? Telling us, e.g., the area of Julian's <http://gosper.org/tripenta.gif> trisectrix (astroid) but not the pentasectrix (quadricuspid epicycloid). Kids: I somehow missed (or completely forgot) the cool integral for the area of the curve enclosed by {x(t),y(t)}: ½∲(x dy - y dx). Supposing a unit radius, what's the area of one of those four "cabochons"? —rwg

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[math-fun] What area is filled by the Dragon joining 0+0i to 1+0i?
by Bill Gosper 10 Nov '19

10 Nov '19

Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research". It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither. You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>) The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg

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[math-fun] Shoelace formula, twin dragon area, Dekking article
by Joerg Arndt 10 Nov '19

10 Nov '19

* Bill Gosper <billgosper(a)gmail.com> [Nov 08. 2019 15:44]: > [...]] > Kids: I somehow missed (or completely forgot) the cool integral for the > area of the curve enclosed by {x(t),y(t)}: ½∲(x dy - y dx). And for the discrete kids, one also has to mention https://en.wikipedia.org/wiki/Shoelace_formula With this one and an automaton for the border of the twin-dragon ( see Example 4.4 in F.\ M.\ Dekking: {Recurrent Sets}, Advances in Mathematics, vol.~44, no.~1, pp.~78-104, (1982). http://dx.doi.org/10.1016/0001-8708(82)90066-4 (now open access) ) you should be able to compute good approximations of the area of the twin-dragon (you asked about it in another email, I seem to recall). Best regards, jj P.S.: Should I feel stupid because I cannot grasp the commutative diagram on page 79? (I really tried!) > [...]

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[math-fun] Elliptic Curve Animation
by Brad Klee 09 Nov '19

09 Nov '19

Plotting around the poles of Edwards's elliptic function psi it is possible to make a fairly nice picture covering 3/4 of the domain [0,2]x[0,2*tau] : https://youtu.be/m5moOyP0zek ( V 1.0, possibly more improvements upcoming) The animation goes through real "a" from ~0 to ~1. When a~0, the central region approaches to the "harmonic hyperboloid", as do four exterior loops when a~1. Today, I finally straightened out how to reduce the exact solution of pendulum equations of motion, at last using the point doubling addition rule to simplify the argument of theta(t) = 2*arcsin(...). Has anyone else done this derivation? Cheers, Brad

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[math-fun] The Sacred Riemann Sphere, Again.
by Brad Klee 07 Nov '19

07 Nov '19

These notes from IPhT are starting to look nice, especially with color figures: https://arxiv.org/pdf/1805.06405.pdf , and middle of page 16 mentions the realization that everyone needs to have about genus zero: "(it has the topology of a cylinder)". Shouldn't this be more than an after-thought? It is another way of saying "both sinh(t) and cosh(t) are singly-periodic functions over the complex plane". This analogy between surface topology and periodicity of parametric functions extends to higher genus, and begets the grand "Riemann Uniformization Theorem". My personal issue, today's gripe, is with Theorem 3.5 "Genus zero = Riemann sphere". This theorem is copied from place to place in just about every reference on algebraic topology. Such practice excludes that "Genus zero = Harmonic Hyperboloid", arguably a more fundamentally correct statement. Students continue to go unaware, while the same mantra gets over-used, again and again. I feel like I have already complained about this, repeatedly. This whole thing about the Riemann Sphere being the one True Form of Genus Zero (TFGZ) has become too dogmatic for me. I'm planning to release more detailed Heresy in the near future, so watch out! Cheers --Brad PS. For more info about "Harmonic Hyperboloid", see also: https://demonstrations.wolfram.com/SemiclassicalApproximationForQuantumHarm…

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[math-fun] Cumulative sums and hits
by Éric Angelini 04 Nov '19

04 Nov '19

Hello Math-Fun, We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process. If, at some stage, the concatenation starts with the string K, we have a hit. K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29 Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34 Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,... HIT! Are all integers potential hits? Best É.

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