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Re: [math-fun] integers that are either square or thrice a square, where do such beasts hide?
by Dan Asimov 30 Aug '18

30 Aug '18

With "rational arguments" — not sure I know what that means. Eisenstein integers should be the ring Z[w] where w = -1/2 + i*sqrt(3/4). Since Z[w] = {K + L*w | K, L in Z} we can compute |K + L*w|^2 = (K - L/2)^2 + (3/4)*L^2 = K^2 - KL + L^2 (which is what I remembered). I'm still in the dark over what rational arguments are, and how K^2 - KL + L^2 squares with what Adam wrote. —Dan Adam Goucher wrote: ----- They're precisely the norms of Eisenstein integers with rational arguments. Wouter Meeussen wrote: ----- > do they occur anywhere naturally? in combinatorics or in number theory? > I bumped into them by accident looking at bracelet-stuff. ----- -----

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[math-fun] Identity for Pi
by françois mendzina essomba2 30 Aug '18

30 Aug '18

Hello, Here is a identity that j found, with a ratio between cosine and hyperbolic cosine. Pi=2*mu*x+2*sum((1/n)*(cos(mu*x)/cosh(alpha*x))^n*cosh(mu*n*x)*sin(alpha*n*x),n=1..infinity); This identity seems true for the following conditions: mu = alpha and x in [1/mu , 2/mu] The value of x, for which this identity is the most convergent is close to : x = (1.5)*(1/mu) The values of x, verifying the identity seem to form a parable whose vertex corresponds to (1/2) * (1 / mu), which I found curious0 Why ???

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Re: [math-fun] Identity for Pi
by Bill Gosper 29 Aug '18

29 Aug '18

Veit, it's obvious for t=π/2. I was able to squeeze 90+digits of π out of t=1, but some of those Borwein frauds can give you thousands: http://www.math.grinnell.edu/~chamberl/courses/444/worksheets/high-precisio… —rwg n 2018-08-29 12:01, Veit Elser wrote: > I suspect it’s true, not fraudulent. > > Try: > > p = 42; > > Simplify[D[ > 2 t + 2 Sum[(Cosh[n t] (Cos[t] Sech[t])^n Sin[n t])/n, {n, p}], {t, > p}] /. t -> Pi/2] > > -Veit > >> On Aug 29, 2018, at 1:14 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> π == 2t + 2Sum[(Cosh[n*t]*(Cos[t]*Sech[t])^n*Sin[n*t])/n, {n,∞}], >> (1≤t<2 ?) >> >> Is anybody playing with this? I find it incredible. Is it one of those >> Borwein or >> Zagier type high precision frauds? Bibasic telescopy? Am I missing >> something >> completely obvious? Inconclusive plotting suggests it holds in a region of >> the >> complex plane shaped like a football silhouette centered at t = π/2 + 0i of >> width > 1 and height < i. (DIY or stay tuned.) >> —rwg >

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Re: [math-fun] Power series question
by Dan Asimov 26 Aug '18

26 Aug '18

A kind soul has pointed out my mismatched parentheses. Aaargh. It should read as follows: ----- Graphing the function f(x) = sgn(x) * |x|^(-1/(x-1)) over the domain (-1, 1) - {0} makes it appear quite smooth around x = 0. Question: Does this function have a power series that converges in some neighborhood of x=0 ? ----- As always: Sorry about that. —Dan ----- >On 25-Aug-18 17:41, Dan Asimov wrote: >> Graphing the function >> >> f(x) = sgn(x) * |x|^(-1/x-1)) >> >> over the domain (-1, 1) - {0} makes it appear quite smooth around x = 0. >> >> Question: Does this function have a power series that converges in >> some neighborhood of x=0 ? -----

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[math-fun] Herschel was an awesome dude, Huygens even greater.
by Bill Gosper 26 Aug '18

26 Aug '18

I just sent this to http://mathworld.wolfram.com/cgi-bin/comment.cgi Herschel: *Comments:*Current text: "A heptomino shaped like the astronomical symbol for Uranus (which was discovered by William Herschel)." Should read: "A heptomino shaped like the astronomical symbol for Saturn (first accurately described by Christiaan Huygens), but regrettably confused with William Herschel, the discoverer of Uranus, by early investigators of Conway's Game of Life. It should be called Huygens, or J heptomino." with a link to Golly's Lexicon: :*Herschel* (stabilizes at time 128) The following pattern which occurs at generation 20 of the B-heptomino. O.. O.O OOO ..O The name is commonly ascribed to the Herschel heptomino's similarity to a planetary symbol. William Herschel discovered Uranus in 1781. However, in point of fact a Herschel bears no particular resemblance to either of the symbols used for Uranus, but does closely resemble the symbol for Saturn. So the appropriate name might actually be "Huygens", but "Herschel" is now universally used by tradition. Herschels are one of the most versatile types of signal in stable circuitry. R-pentominoes and B-heptominoes naturally evolve into Herschels, and converters have also been found that change pi-heptominoes and several other signal types into Herschels, and vice versa. See elementary conduit. On 28 April 2015, Michael Simkin found the smallest known glider gun--a lone Huygens plus six Lazarushian blocks, period 120. ............o...........oo.....oo ............ooo.........oo.....oo ............o.o.................. ..............o............oo.... ...........................oo.... ................................. ................................. ................................. ....oo........................... ....oo........................... ................................. oo.....oo........................ oo.....oo........................ --rwg

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[math-fun] Power series question
by Dan Asimov 25 Aug '18

25 Aug '18

Graphing the function f(x) = sgn(x) * |x|^(-1/x-1)) over the domain (-1, 1) - {0} makes it appear quite smooth around x = 0. Question: Does this function have a power series that converges in some neighborhood of x=0 ? —Dan

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[math-fun] Fwd: a huge boon to the dirigible industry
by Bill Gosper 25 Aug '18

25 Aug '18

(Re-sent due to xmission.com braindamage.)---------- Forwarded message --------- From: Bill Gosper <billgosper(a)gmail.com> Date: Sat, Aug 18, 2018 at 9:36 AM Subject: a huge boon to the dirigible industry To: Wolfram Technical Support <support(a)wolfram.com> In[31]:= ChemicalData["Nitrogen", "MolecularMass"] Out[31]= Quantity[28.014, "AtomicMassUnit"] (A mole of nitrogen gas weighs ~one ounce.) In[32]:= ChemicalData["Air", "MolecularMass"] Out[32]= Quantity[794.9, "AtomicMassUnit"] Takes my breath away. --Bill

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Re: [math-fun] Fwd: beautiful visualization of the first million integers
by Dan Asimov 23 Aug '18

23 Aug '18

P.S. One commenter on the github Python code page for these pictures wrote: ----- It's worth noting that any structure that is not an algorithmic artificat is really the frontier of numbers at some threshold (in this case, 1 million). Every possible finite binary vector corresponds to at least one integer in this representation (actually, infinitely many, as all repeated factors are contracted onto a single binary indicator). Therefore, in the infinite set of integers, all points are present, at every possible angle or distance from each other, and the space would be perfectly evenly filled. However, by cutting off at some specific n, we see the “surface” that we have reached by composing together primes such that their product is less than n. ----- Seems right to me! —Dan

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Re: [math-fun] Fwd: beautiful visualization of the first million integers
by Dan Asimov 23 Aug '18

23 Aug '18

----- the inexplicable decision to throw away all information about the exponents except whether they are zero or nonzero. ----- I just want to comment on this description, not on the visualization per se. First of all, there is no question that the quantity of information discarded is vast. But there is still a vast amount of information retained. Many wonderful mathematical discoveries have been made by forgetting a lot of information. A detailed metric space could be forgotten into a mere topological space, which could be forgotten into its homotopy groups, say (these are typically finitely generated groups {π_n}, one group for each positive integer with all but the first one abelian). And in this case, we can see identifying all positive powers of a prime factor with each other as representing — if not all integers — instead, all possible *divisibilities*: Each integer N is divisible by certain primes and not by others. Each given pattern of divisibilities is represented once and only once in this reduced information. Okay, so that's what it's actually a picture of. —Dan

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[math-fun] Fwd: beautiful visualization of the first million integers
by Brent Meeker 23 Aug '18

23 Aug '18

A fascinating display of mathematical fireworks. Brent -------- Forwarded Message -------- Each integer is represented as a vector of its prime factors, in a high-dimensionality representation such that: [1 0 1 0 0 0 0 ...] means that the number has factors 2 and 5 (first and third primes). Then they apply a dimensionality reduction algorithm to transform these vectors into 2D vectors with real-valued coordinates. I think the result is beautiful: https://johnhw.github.io/umap_primes/index.md.html Cheers

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