February 2020 - mathfun@mailman.xmission.com

[math-fun] Morley's Theorem
by James Propp 16 Feb '20

16 Feb '20

Is Donald J. Newman's proof of Morley's Theorem the same as Conway's? The article https://www.researchgate.net/publication/267921593_Is_John_Conway's_Proof_of_Morley's_Theorem_the_Simplest_and_Free_of_A_Deus_Ex_Machina references Newman's article (reference [9]) but never discusses it, and from what I can see at https://www.cut-the-knot.org/triangle/Morley/newman.shtml it seems to me that the key insight is the same in both cases. Have any of you studied Morley's Theorem and its proofs in detail, and feel qualified to weigh in on this? There's more good stuff at https://www.cut-the-knot.org/ctk/MorleysRedux.shtml than I can process... Thanks, Jim Propp

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Re: [math-fun] a long series of primes
by Keith F. Lynch 15 Feb '20

15 Feb '20

Mike Stay <metaweta(a)gmail.com> wrote: > Simon Plouffe <simon.plouffe(a)gmail.com> wrote: >> Is there simpler method to generate any number of primes ? > "Simplicity" depends entirely on the programming language. In > Mathematica, the simplest way is Prime[n]. One might think that > there are "natural" universal Turing machines, in the sense that > unnatural ones can simulate natural ones with short interpreters, > but not vice-versa. Somewhat surprisingly, that turns out to > be false. Cite? Also, from context I think Simon really means a faster way, not a simpler way. There's often a trade-off between the speed and the complexity of an algorithm. And how difficult it is to write a program often depends on how well the programming language hides the complexity. Just because it's easy to write Factor(1+2^3^4^5^6^7^8^9) doesn't mean you'll get an answer before the heat death of the universe. Also, physically building Turing-equivalent machines requires physical components, such as transistors, gears, or electromechanical relays. Some tasks inherently require either more such components, or that the fixed number of such components change state more times, than other tasks do.

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[math-fun] Quanta: "The Map of Mathematics"
by Brad Klee 15 Feb '20

15 Feb '20

See: https://www.quantamagazine.org/the-map-of-mathematics-20200213/ It's no surprise that scientists would eventually get to the idea of making a map of scientific ideas. Such a task could be useful for orienteering, and possibly an opportunity to explore Hofstadter-esque quine-making. Such creations can eventually be seen as most indicative of the psychology particular to the cartographer or cartographers. Often times you will get some sort of phylogenetic tree structure, which needs to be amended with extra connections. The necessity of amendments seems to be the only common feature between the various tree models, and this underscores the empirical fact that mathematical ideas tend to converge as often as they diverge (ideas are not ultimately constrained by sexual or genetic reproduction). Alexandru Buium did not make the same mistake with definite / indefinite articles when producing "a map of mathematics", see also: https://math.unm.edu/~buium/mapp.pdf These are both interesting takes, but not entirely similar to the map that I would probably prefer, because time is not included as a variable of the branching process. Is there any reason for "Function Phylogeny"? And what is the best way to limit subjectivity? Any thoughts in general? --Brad

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[math-fun] Pennies in a Frying Pan
by Richard 14 Feb '20

14 Feb '20

Dear Funsters Imagine n pennies in a frying pan. They are all flipped randomly at once and those landing heads are removed and placed heads up on a flat surface. The process is repeated with the remaining pennies until all coins show heads. What is the expected number of flips, f(n), for n coins? f(1) = 2, f(2) = 8/3, f(3) = 66/3/7, f(4) = 1104/3/7/15, f(5) = 37050/3/7/15/31 etc. The numerators follow the sequence 2, 8, 66, 1104, 37050, 2482200, 31336530, 88081529760, 46640940221610.... 1. I don't find this sequence in the OEIS. Should it be there or is there a related sequence that is there? 2 f(64) = 7.34399... f(128) = 8.33837... f(256) = 9.33556... f(512) = 10.33415... f(1024) = 11.33345... f(2048) = 12.33309... f(4096) = 13.33292... f(8192) = 14.33283... f(16384) = 15.3327... Perhaps someone can verify these results and determine the limit of e(k) as k approaches infinity for f(2^k) = k+1+e(k). Any help is appreciated. Dick Hess

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[math-fun] a long series of primes
by Simon Plouffe 13 Feb '20

13 Feb '20

Hello, I have been working on ways to generate primes with the simplest formula. Here is one : if c = 31622.7 ... then { c^n } is prime for n = 2 to 388 : 387 consecutive primes. { } is the nearest integer. The sequence begins like this : 1000000007 31622776952311 1000000014783746303 31622777186062677745609 1000000022175619536498921059 31622777419814234539614807614633 ... and not listed in the OEIS. Here is the complete list and more precise value of the constant. http://plouffe.fr/NEW/List%20of%20387%20consecutive%20primes.txt I developed a method based on Monte-Carlo and Simulated Annealing https://en.wikipedia.org/wiki/Simulated_annealing The method allows me to find other candidates which would give a longer series of primes. This formulation is the simplest I could imagine. Is there simpler method to generate any number of primes ? Best regards, Simon Plouffe

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[math-fun] Sparse rulers proof
by ed pegg 12 Feb '20

12 Feb '20

TLDR version: ||||.......................|....|...|...|...|...|...|..|..| Sparse rulers allow the measuring of all distances up to length n (58) with a minimal number of marks M (13). The author believes B. Wichmann almost completely solved the problem in 1963. From Wichmann's sparse ruler recipe, M_n - ⌈√(3 n+9/4)⌋ can be shown to usually be 0 or 1 (the excess). The length 58 ruler at the start of this paragraph is conjectured to be the longest sparse ruler with fewer marks than a Wichmann construction, but no proof is known. Adding a single mark to the end of Wichmann's construction allows the building of excess-01 rulers for all lengths over 257992. The author's code found solutions for hundreds of difficult smaller lengths such as 1792, 5657 and 16617 to establish that excess-01 rulers always exist. The optimality of excess 1 solutions for lengths 474, 501, 582, 609, 669, 792, 793, ... may be doubtful since they break the excess pattern. The OEIS version is https://oeis.org/A326499 https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bou… --Ed Pegg Jr

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[math-fun] Suggestion for next Iowa Basic Skills Test
by Brad Klee 11 Feb '20

11 Feb '20

Did anyone else have to take these tests as a kid? There was no real reward for doing well, but if you fail, you get labeled as "not-up-to-standard". I can't remember any of their problems, but thought of something else today: An election has five candidates finish with double-digit percentages, though, all less than 30%. After about 20 hours, the committee assigned to count votes decides at first to announce only 62% of all results. Define an "exclusion scenario" as a set C_1, C_2, ..., C_n of candidates who are announced to have zero percent support on a fair tally of the 62% subset. How many different exclusion scenarios are possible when allowing for any 62% subset? Is it possible to completely exclude the overall winner? I think this can be solved using standard combinatorics, but I am wondering if anyone would understand the implications. --Brad

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Re: [math-fun] Mystery tables
by Keith F. Lynch 10 Feb '20

10 Feb '20

Gareth McCaughan <gareth.mccaughan(a)pobox.com> wrote: > Keith F. Lynch wrote: >> Most such values of x are transcendental numbers. But not all. >> What rational values can x have? Can it have an irrational >> algebraic value? > (This is values of x where m^x - 2n^x = 1, with m,n positive > integers.) Right, i.e. where n is the xth power mean of 1 and m. My first table. > I betcha x is never an irrational algebraic number, and I betcha > there will not in our lifetimes be sufficient mathematical > technology to prove this. Maybe so, but I've made some progress on rational values of x. It's 0 whenever m is the square of n, 1 whenever m+1 is 2n, and 2 whenever m^2+1 = 2n^2, i.e. where m and n are corresponding elements of A002315 and A001653. For instance 7,5 and 41,29. There are infinitely many solutions for each of x=0, 1, and 2. It would also be m^x+1 = 2n^x for other integers x, but according to http://www.math.mcgill.ca/darmon/pub/Articles/Research/18.Merel/paper.pdf (equation 1) there are no such integers. It's 1/j whenever there are integers a and j such that m = a^j and n = ((a+1)/2)^j. For instance 9,4 for 1/2 and 27,8 for 1/3. There are infinitely many solutions for every positive integer j. It's 2/j whenever j is odd and m and n are corresponding jth powers of A002315 and A001653, for instance 343,125, i.e. 7^3, 5^3. There are infinitely many solutions for every positive integer j. (For even j, the solution for 2/j is that for 1/(j/2).) I can't find any other rational values of x. Can you? >> Hence my second table. I assume they're all transcendental except >> of course where x=y, in which case z=x=y. Can anyone prove this? >> Or find a counterexample? > (This is values of x where m^x + n^x = 2x^x, with m,n positive > integers.) > I betcha x is never algebraic unless m=n, and I betcha there will not > in our lifetimes be sufficient mathematical technology to prove this. If they are irrational, they are always transcendental. See the Gelfond-Schneider theorem, also known as Hilbert's 7th problem. That leaves the remote possibility that there's some non-integer rational x.

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Re: [math-fun] Mystery tables
by Keith F. Lynch 09 Feb '20

09 Feb '20

Steve Witham <sw(a)tiac.net> wrote: > "Keith F. Lynch" <kfl(a)KeithLynch.net> >> -1: Harmonic mean > Kewl, now I think I can average two resistors. That depends on whether the resistors are in series or parallel. And on whether you're averaging their resistances or their conductances. Harmonic means show up in lots of different contexts. This morning, I happened to read a blog comment dated just three days ago, which said, in part: Or there was this think PISA question "We traveled to a city at a speed of 20km/h and traveled back at 30 km/h, what was our average speed for the whole journey?" And very few people get it right and not because they are stupid but because they do not use pen and paper to work it out. Why? If you just go "assume they are 100km away" or "assume the trip there was one hour" and work it out on paper there is no way you get it wrong. Seriously. I don't know why people don't do that. http://esr.ibiblio.org/?p=8558 The correct answer is of course the harmonic mean of 20 and 30, which is 24. I solved it my head with no pen, paper, assumed distance, or assumed duration. >> 0: Geometric mean > I got as far as wondering whether this was something like what > happens when integrating x^(-1)... instead of getting (x^0) / 0, > and then went to wikipedia... which treats it like a convention: > Do these means change smoothly as x crosses zero? Yes. For what it's worth, my program which generated my first table gave the correct answer, 0.000000, for 4,2, 9,3, 16,4, etc., without treating them as special cases. (Okay, some of them gave -0.000000, and I "cheated" by manually deleting the minus sign before posting.) Note that I carefully defined it as (a^x + b^x) / 2 = c^x, not as c = the xth root of ((a^x + b^x) / 2). Trying to take the zeroth root of a number is problematic. Raising a number to the zeroth power is not. My program sets b to 1, then for each pair of positive integers c<a<1000 tries various values of x, quickly converging on the solution with a binary search. It takes less than a minute to fill the table, which has about half a million entries. Related problem: What if everything is an integer? I've found cases where there are positive integers a<b such that there are three solutions to (a^x + b^x) / 2 = c^x in which c and x are both integers. I haven't found any where there are four such solutions. Can you find one or prove there aren't any? Another related problem: Find an infinite integer sequence in which no two terms have an integer power mean in the sequence. For instance A005836 is an infinite integer sequence in which no two terms have an arithmetic mean in the sequence. I want the same thing but also for geometric means, harmonic means, quadratic means, etc., all at the same time.

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[math-fun] Oh No! Malus's law seems to fail...
by Brad Klee 08 Feb '20

08 Feb '20

See: https://0x0.st/ii4o.jpg for a fun experiment, and: https://www.exploratorium.edu/snacks/polarized-light-mosaic for more explanation. --Brad

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