April 2019 - mathfun@mailman.xmission.com

Re: [math-fun] A few questions about prime generating functions
by Dan Asimov 29 Apr '19

29 Apr '19

That Wikipedia article Formulas for primes contains this interesting tidbit: ----- It is not even known whether there exists a univariate polynomial of degree at least 2, that assumes an infinite number of values that are prime; ----- The article appears to be referring to *integer* polynomials. What about a *real* polynomial P(x), deg(P) >= 2 ??? Can some such P(x) be proved to represent infinitely many primes? —Dan ----- II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? -----

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[math-fun] A few questions about prime generating functions
by Dan Asimov 29 Apr '19

29 Apr '19

I. It's widely believed that there's no nice function f : Z+ —> Z+ whose values are all prime: f(Z+) ⊂ Primes , where Primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...}. Are there any actual theorems that say as much — necessarily defining "nice" ? Or at least restricting the classes of nice functions for which this might be possible? How about heuristics indicating why this should be hard? * * * II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? Dirichlet's famous 1837 theorem states that every arithmetic sequence of form {X_n = A + B n}, where A, B are relatively prime integers, contains infinitely many primes.* But I don't know of any other cases. Is that because they aren't known? —Dan ————— * I once read the proof (as a very readable chapter in Elliptic Curves by Anthony Knapp) and thought it astonishing innovative, but especially for that long ago!

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[math-fun] Hot tea sucks
by Bill Gosper 29 Apr '19

29 Apr '19

I *thought* Tom Knight beat Brent Meeker with the top it up solution, but that mail seems only intermittently visible in my RoundCube. On 2019-04-26 09:05, Veit Elser wrote: > First hike up Mt. Umunhum (always wanted to use that in a sentence) or > someplace higher. > You've touched on an interesting subject which I encourage you to Google further. Les Earnest claims the station up there was part of the SAGE air defense system, which was the biggest taxpayer fraud in history. But he also mentions that, 20 miles away, it was blitzing the Stanford AILab computer at 13 second intervals. I.e., whatever they were doing, the people up there were serious. The actual radar building is improbably large: http://www.redwoodhikes.com/Skyline/MtUmunhum1.jpg There were people actually living there year-round. They even had a bowling alley. (I bet that, summer afternoons, it's *way* too windy for volleyball!) And for years after it closed, the area was very off-limits to hikers and motorists. —rwg >> On Apr 26, 2019, at 11:12 AM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >> I brew it in a 46oz pickle jar that easily tips over, and leaks if the lid >> isn't snug. >> But if I forget to momentarily loosen the lid within a few minutes, a >> powerful vacuum >> develops that makes it virtually impossible to unscrew. What is the >> trivial solution? >> —BillI

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Re: [math-fun] Hot tea sucks.
by Bill Gosper 29 Apr '19

29 Apr '19

On 2019-04-26 21:17, Andres Valloud wrote: > Does it involve one of those lids that snaps? No, just a wide-mouth twist off Mezzetta's jar. > You could put in the lid > with the snapper up, and when it snaps down then you know you have to > undo the lid. > > I suppose you could also forget you were brewing tea, so the snapper > sound won't be enough to remind you. But perhaps the lid snapper is > enough to keep the strong vacuum from forming in the first place. > > Filling the jar completely [Brent Meeker's suggestion] has the problem >that the jar will very likely > spill while closing as well as opening, and it sounds like that's > something you want to avoid. Exactly. > So, almost fill it, and use a snapping lid. Recall that my (satisfactory) solution of merely blowing cool air under the lid as you replace it the first time. How robust is Brent's top-up suggestion? I purposely underfilled by about .25", and closed it quickly without blowing away the "steam", waiting extra long before trying the lid. It came off! Grudgingly. Believing the experiment to be over, a chugged several swallows, since it wasn't very hot. Then I reclosed it and forgot about it for maybe an hour. Next time, it was hard as heck to get off! The dT was modest enough, but the dV wasn't! Double verification for Brent's volume emphasis. —rwg > > On 4/26/19 15:54 , Bill Gosper wrote: >> I BCCed a lot of people and got a lot of nontrivial suggestions. >> (Restaurateur) Mike Wang's rubber band suggestion >> was probably best. Just encircle the lid with a big one from a broccoli >> bunch. It's amazing if you've never tried it. >> But that's not trivial enough! (I forget where I stashed my rubber band.) >> Someone suggested that if I have a mountain >> handy, I could drive up it. Nobody suggested the dicey solution of >> sticking the jar back in the 𝜇wave. Really hot tapwater >> should work, but our water heater is set way low. C'mon people—*trivial*! >> >> SPOILER: >> In[532]:= StringReverse@"!no ti pals uoy sa dil eht rednu ria )looc >> ylevitaler( fo llufgnul kciuq a wolb tsuJ" >> —rwg >> >> On Fri, Apr 26, 2019 at 8:12 AM Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> I brew it in a 46oz pickle jar that easily tips over, and leaks if the lid >>> isn't snug. >>> But if I forget to momentarily loosen the lid within a few minutes, a >>> powerful vacuum >>> develops that makes it virtually impossible to unscrew. What is the >>> trivial solution? >>> —Bill

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Re: [math-fun] math of carnival games
by Dave Dyer 27 Apr '19

27 Apr '19

This is a very entertaining video illustrating the power of statistics used as a carnival scam. https://www.youtube.com/watch?v=527F51qTcTg&feature=youtu.be&fbclid=IwAR0R3…

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Re: [math-fun] math of carnival games
by Dave Dyer 27 Apr '19

27 Apr '19

This is a very entertaining video illustrating the power of statistics used as a carnival scam. https://www.youtube.com/watch?v=527F51qTcTg&feature=youtu.be&fbclid=IwAR0R3…

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Re: [math-fun] Neptune-Pluto orbital resonance
by Keith F. Lynch 27 Apr '19

27 Apr '19

Bill Gosper <billgosper(a)gmail.com> wrote: > Wikipedia: > Under some circumstances, a resonant system can be stable and > self-correcting, so that the bodies remain in resonance. I'm reminded of one of my long term plans. One of my *very* long term plans. The sun will gradually become a red giant and incinerate our planet. To save Earth, it will be necessary to gradually move it into a larger orbit for the duration. And then into a smaller orbit once the sun becomes a white dwarf. There are at least three main ways Earth might be moved outwards: * Loop asteroids around it and Jupiter, pulling Earth up and pulling Jupiter down. (Of course this has problems once they approach each other.) * Place a giant mirror near Earth, such that sunlight will push the mirror forwards in Earth's orbit, and Earth's gravity will make Earth follow it. * Launch lots of stuff from Earth backwards along its orbit. Fortunately, we have time. We can spend the next hundred million years ignoring the problem, the hundred million years after that thinking about it, and the hundred million years after than gradually accelerating the Earth outwards such that at the end of that hundred million years its outward drift reaches the speed of a snail. However, it's not clear whether on its outward journey it would hit any bad resonances with the other planets that might cause planets to collide with each other or with the sun, or expel a planet from the solar system. (Of course once Earth reaches Mars's orbit, a collision is inevitable.) The obvious solution, at least with Mars, is to move the other planet outwards too. They will all move outwards even without our intervention, due to the sun losing mass to radiation and to the solar wind. Not enough to save us, unfortunately. Nor should it destabilize the solar system. If they retain their orbital velocity (and why wouldn't they?) the orbital radii and the orbital periods should all increase by the same proportion that the sun loses mass. (Kepler's law that says the square of the period is proportional to the cube of the radius doesn't apply here, as it implicitly assumes the sun's mass is unchanging.) I'll give details of my reasoning on request. Kepler's law most certainly *does* apply if we move all the planets outwards ourselves. So we can't just enlarge all the orbits keeping the proportions of their orbital radii constant. Can we get away with enlarging all the orbits keeping the proportions of their orbital *periods* constant? And is there a cheaper way, in which we can leave most of the larger planets alone? Has anyone studied this problem? Is there any known way to study it other than a massive amount of numerical simulation, i.e. a digital orrery? Urgent! Please reply within the next quarter billion years! Thanks.

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[math-fun] Neptune-Pluto orbital resonance
by Bill Gosper 27 Apr '19

27 Apr '19

Wikipedia: Under some circumstances, a resonant system can be stable and self-correcting, so that the bodies remain in resonance. Examples are the 1:2:4 resonance of Jupiter <https://en.wikipedia.org/wiki/Jupiter>'s moons Ganymede <https://en.wikipedia.org/wiki/Ganymede_(moon)>, Europa <https://en.wikipedia.org/wiki/Europa_(moon)> and Io <https://en.wikipedia.org/wiki/Io_(moon)>, and the 2:3 resonance between Pluto <https://en.wikipedia.org/wiki/Pluto> and Neptune <https://en.wikipedia.org/wiki/Neptune>. In[234]:= MinorPlanetData["Pluto"][EntityProperty["MinorPlanet", "OrbitPeriod"]] Out[234]= Quantity[247.92065, "JulianYears"] In[235]:= PlanetData["Neptune"][EntityProperty["Planet", "OrbitPeriod"]] Out[235]= Quantity[164.79132, "JulianYears"] In[236]:= ContinuedFraction[%%/%] Out[236]= {1, 1, 1, 55, 1, 1, 1, 7} That's a fairly lousy approximation to 2:3. What's going on? —rwg

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Re: [math-fun] Should I be surprised?
by Keith F. Lynch 27 Apr '19

27 Apr '19

James Propp <jamespropp(a)gmail.com> wrote: > The 91st Fibonacci number is > 4,660,046,610,375,530,309 > Should I be surprised that the first eight digits are even? > Should I be surprised that, if you strip out the zero-digits, this > number consists of six even digits followed by eight odd digits? No. This is called the Texas Sharpshooter Fallacy. The a priori odds of that exact pattern are small, but it's taken from vast population of equally unlikely patterns, e.g. that in base 17 the digits of the 23rd Catalan number are in ascending sequence if you exclude the digit "3." (I just made that up; I haven't checked. If by remarkable coincidence that actually *is* true, that's *not* an example of the fallacy, since I "claimed" it before checking.)

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[math-fun] Should I be surprised?
by James Propp 27 Apr '19

27 Apr '19

The 91st Fibonacci number is 4,660,046,610,375,530,309 Should I be surprised that the first eight digits are even? Should I be surprised that, if you strip out the zero-digits, this number consists of six even digits followed by eight odd digits? I know that the initial digits of the Fibonacci numbers have an asymptotic equidistribution property, but 91 is not that big... Jim Propp

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