November 2013 - mathfun@mailman.xmission.com

[math-fun] (no subject)
by Bill Gosper 13 Nov '13

13 Nov '13

How about Hilbert's squarefill evaluated with a fixed, irrational step size? gosper.org/hilbrand.png --rwg BMeeker>How about lattice points plus small random vectors. Brent On 11/12/2013 9:16 PM, Warren D Smith wrote: WDS>In the plane (or some other low dimensional space) I want a good infinite point set that has "density 1." What does "good" mean? Well, "more uniform than random" in some vague sense, such as sphere packing, sphere covering, and providing good estimates of integrals via sums. BUT, what I also want, is the set has no repeated distances. So lattices are undesirable. Random point sets will not repeat a distance (with probability 1) so they are great, except they fail to be good in the sense of the previous paragraph, while lattices can be quite good in that sense. Another thing I want is an easy, algorithmic description of the point set, and it should be deterministic. The question is how to get the best of both worlds. I admit, I am being vague here. Ideas are solicited.

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[math-fun] good deterministic algorithmic point sets without repeated distances?
by Warren D Smith 12 Nov '13

12 Nov '13

In the plane (or some other low dimensional space) I want a good infinite point set that has "density 1." What does "good" mean? Well, "more uniform than random" in some vague sense, such as sphere packing, sphere covering, and providing good estimates of integrals via sums. BUT, what I also want, is the set has no repeated distances. So lattices are undesirable. Random point sets will not repeat a distance (with probability 1) so they are great, except they fail to be good in the sense of the previous paragraph, while lattices can be quite good in that sense. Another thing I want is an easy, algorithmic description of the point set, and it should be deterministic. The question is how to get the best of both worlds. I admit, I am being vague here. Ideas are solicited.

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Re: [math-fun] density of primorial+1 primes
by Bill Gosper 12 Nov '13

12 Nov '13

RCS> I'll guess that the False value for a_20 is from some small factor, and that the Aborted values are because the test ran out of time before completing even one superimposed test. Try asking for a few digits of 2^(a_12) (mod a_12), and then see how far you can get with a_13 ... before running out of gas. Rich --------- I lose; you win: In[11]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 12] Out[11]= 3366937412175501591717367777101460820640224690509113100163277\ 6539067634240985481258893779792671452248127456725155260727280357834657\ 6076738692420210769091266874710826881662611178920169012031118635505607\ 4310982786125119105965083788824019629431580952133240830246962982908185\ 9746496798634116562354568928408751636377986351524509214103104546223261\ 2396128362132130189597314029272968730759961945103880790832377276137373\ 9360515945586462465327970571098621823207535551580401784234812938313843\ 6459926693612089970288170337138209267562435148320741084111322614415098\ 1287168301351535936792810718614137830956322091599451719533327897728719\ 2022822793035110202282332960887794400493112644185559756058506634629406\ 8091290492788783650041163770989366969890285626688901808066658454549560\ 7443533211603079106672889774858913012113951640476740718704380411099010\ 204 In[13]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 6] Out[13]= 4457343787994 (Note: a_0:=2 .) In[14]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[14]= 2 In[16]:= PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 4] Out[16]= 1103 In[17]:= PowerMod[3, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[17]= 3 In[18]:= PowerMod[8, #, #] &@Nest[#^2 - # + 1 &, 2, 5] Out[18]= 8 In[12]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 13]]] Out[12]= {0.105467, 2.415419924381151*10^1667} In[19]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 14]]] Out[19]= {0.647739, 7.344539857481122*10^3334} In[20]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 15]]] Out[20]= {3.835341, 6.597333832436021*10^6669} In[21]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 16]]] Out[21]= {21.430643, 1.412908137918137*10^13339} In[22]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 17]]] Out[22]= {119.182441, 8.856579016766857*10^26679} In[23]:= Timing[N[PowerMod[2, #, #] &@Nest[#^2 - # + 1 &, 2, 18]]] Out[23]= {576.777472, 7.425006686698617*10^53359} In[24]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 18]]] Out[24]= {577.039477, False} In[25]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 19]]] should take about an hour. Meanwhile, we can now conjecture, as with Fermat #s, that there are no more of these primes! --rwg Quoting Bill Gosper <billgosper(a)gmail.com>: <eliding earlier comments --rich> Only slightly less infeasible might be the coprime sequence A000058 (interesting comments). In[10]:= NestList[#^2 - # + 1 &, 2, 9] Out[10]= {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443} In[8]:= N[Timing[Nest[#^2 - # + 1 &, 2, 21]]] Out[8]= {0.007302, 1.853733657356281*10^426880} (Half a million digits in 7ms?) In[6]:= TimeConstrained[PrimeQ[#], 1] & /@ NestList[#^2 - # + 1 &, 2, 21] Out[6]= {True, True, True, True, False, True, False, False, False, False, False, False, False, False, False, False, False, False, $Aborted, $Aborted, False, $Aborted} where $Aborted almost certainly means True. Changing the timeout from 1 to 288 sec did zilch. Decertifying a_20 In[9]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 20]]] Out[9]= {0.150525, False} took < 1/6 sec. Can we show there are no (strong?) pseudoprimes of this form? --rwg SUPERIMPOSED PSEUDOPRIMES <disformatting earlier comments --Firefox>

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Re: [math-fun] density of primorial+1 primes
by Bill Gosper 12 Nov '13

12 Nov '13

DanA> I'm curious: Can one state in closed form what is the asymptotic density of prime numbers among numbers Q(n) of the form Q(n) := primorial(n) + 1 i.e., Q(n) = 2*3*5*7*...*p_n + 1 (where p_n is the nth prime) ??? CG>It's not even known that there are infinitely many. Caldwell & Gallot conjecture that there are about e^gamma * log x prime Q(n) with n <= x, that is, that the usual heuristic gives the right answer. I think sieve theory can give an upper bound if that's enough for your purposes. Charles Greathouse WDS>--I would think that Q(n) is prime with "probability" about loglog(Q(n)) times greater than the usual 1/log(Q(n)) probability. [Warren] Only slightly less infeasible might be the coprime sequence A000058 (interesting comments). In[10]:= NestList[#^2 - # + 1 &, 2, 9] Out[10]= {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443} In[8]:= N[Timing[Nest[#^2 - # + 1 &, 2, 21]]] Out[8]= {0.007302, 1.853733657356281*10^426880} (Half a million digits in 7ms?) In[6]:= TimeConstrained[PrimeQ[#], 1] & /@ NestList[#^2 - # + 1 &, 2, 21] Out[6]= {True, True, True, True, False, True, False, False, False, False, False, False, False, False, False, False, False, False, $Aborted, $Aborted, False, $Aborted} where $Aborted almost certainly means True. Changing the timeout from 1 to 288 sec did zilch. Decertifying a_20 In[9]:= Timing[PrimeQ[Nest[#^2 - # + 1 &, 2, 20]]] Out[9]= {0.150525, False} took < 1/6 sec. Can we show there are no (strong?) pseudoprimes of this form? --rwg SUPERIMPOSED PSEUDOPRIMES

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Re: [math-fun] math-fun Digest, Vol 129, Issue 12
by Warren D Smith 11 Nov '13

11 Nov '13

>> I'm curious: Can one state in closed form what is the asymptotic density >> of prime numbers among numbers Q(n) of the form >> >> Q(n) := primorial(n) + 1 >> >> i.e., >> >> Q(n) = 2*3*5*7*...*p_n + 1 >> >> (where p_n is the nth prime) ??? --I would think that Q(n) is prime with "probability" about loglog(Q(n)) times greater than the usual 1/log(Q(n)) probability.

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[math-fun] density of primorial+1 primes
by Dan Asimov 11 Nov '13

11 Nov '13

I'm curious: Can one state in closed form what is the asymptotic density of prime numbers among numbers Q(n) of the form Q(n) := primorial(n) + 1 i.e., Q(n) = 2*3*5*7*...*p_n + 1 (where p_n is the nth prime) ??? --Dan

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Re: [math-fun] circles, ellipses, foci
by Bill Gosper 11 Nov '13

11 Nov '13

Yes. What a beautiful and ingenious way to prove the counterintuitive fact that the intersection of a plane and a cone is an ellipse (generically)! --Dan On 2013-11-11, at 9:29 AM, rkg wrote: Does everyone know about the Dandelin spheres? A plane intersects a cone in a conic section. The spheres inscribed in the cone and touching the plane do so at the foci. For proof, note that the tangents to a sphere from a point are equal in length, and use the `pins and string' construction for the ellipse (or hyperbola, or parabola -- focus-directrix for this last.) R. On Mon, 11 Nov 2013, Henry Baker wrote: Take a circle & look at it along its axis; it appears as a circle. Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse. Consider the foci of the ellipse. Is there anything interesting and/or cool about the relationship of the foci and the angle alpha? (I don't know any interesting answer; I'm just curious.) Aha. http://en.wikipedia.org/wiki/Dandelin_spheres . Thank you Richard. Here's some recent mail to some youngsters complaining that http://en.wikipedia.org/wiki/Hyperbola doesn't seem to clearly connect the conic section definition to the distances-to-foci definition. http://www.tweedledum.com/rwg/hyperb.GIF is an old Macsyma picture I made. At greater length, jam two (possibly unequal) spheres into opposite branches of a cone, tangent to it in rings of distance r1 and r2 from the apex. A plane tangent to both spheres cuts the cone in a hyperbola whose foci are those tangencies. >From the diagram, the distance of any point on the hyperbola from the farther focus, minus the distance from the nearer focus, will always be r1+r2. --Bill Just saw a thm by Bruce Reznick: If a triangle's vertices are (square) gridpoints, but its sides intersect no other, and its interior contains just one, then that point is the centroid. I wonder if this could be true for "If the sides and interior of a triangle each contain one gridpoint,..." (Quickly shot down by Josh Pollock.) In a message vaporized by GMail, I also mentioned (reconstruction:) Here's one not even Coxeter knew (until he stomped off to the U. Toronto library and found http://archive.org/stream/atreatiseonanal00rogegoog#page/n169/mode/2up .) Thm: The locus of the apex ("vertex") of a cone containing a fixed ellipse (in 3-space) is a hyperbola through the foci of that ellipse, and the locus of the apex of the cone through that hyperbola is the original ellipse, which pierces the foci of the hyperbola. http://gosper.org/conethm.png is a fairly lame illustration--a cross section showing two cones from each family. With translucency, it might make a nice Mma animation, with the two conics fixed in space and the two conesweeps alternating. We could smoothly switch branches of the hyperbola when the cone thru the ellipse goes planar. --rwg

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[math-fun] circles, ellipses, foci
by Henry Baker 11 Nov '13

11 Nov '13

Take a circle & look at it along its axis; it appears as a circle. Now tilt the axis of the circle at an angle alpha; it now appears as an ellipse. Consider the foci of the ellipse. Is there anything interesting and/or cool about the relationship of the foci and the angle alpha? (I don't know any interesting answer; I'm just curious.)

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[math-fun] graphs with E edges
by Warren D Smith 11 Nov '13

11 Nov '13

CLAIM: The number of (isomorphism classes of) connected graphs -- indeed even merely the number of Eulerian graphs -- with E edges exceeds X^X where X=[1-o(1)]*E when E-->infinity. Also it is upper bounded by same, but now with X=[1+o(1)]*E. Also all this is valid for multigraphs (meaning you are allowed to have more than one edge joining two particular vertices). I recently proved this while doing something else. I have not seen it before but calling it "a new result in graph theory" is probably unjustified since proofs are easy enough that most anybody would probably rederive it when they needed it (?). Anyhow it can be proven in 1 page, and the proof is by no means unique.

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Re: [math-fun] What is the tennis-ball curve?
by Bill Gosper 10 Nov '13

10 Nov '13

On 2013-11-09 12:38, Fred Lunnon wrote: > Good point about the stretch issue. Increasing the radius r makes the > Enneper-sphere curve progressively "waisted" in a more traditional fashion. > Via symmetry, the angular distance across the boundary at the inflections > must always equal pi/2 ; so perhaps the optimal curve should have > distance pi/2 across the waist as well, whence > r^2 - 5 sqrt2 r + 2 = 0 , r = 0.2951635567... > See > https://www.dropbox.com/s/7pi27klhpdt6yxk/tennis_ball_2952.jpg > > I'm afraid I do not understand how Bill's onepiececover.png maps > to tennis.gif : Neither do I. Try gosper.org/tennis2.gif ? >surely the semicircular ends of the strip should be > replaced by copies of the central section, bisected by a line across > the cusps? > > Fred Lunnon > Brent M> An interesting idea (although novelty isn't what you want in a tennis > ball). The symmetry is different so it might have an[ e]ffect on the > aerodynamics depending on the spin axis. > > Brent Meeker If anybody complains, just add more fuzz. --rwg > On 11/9/13, Bill Gosper <billgosper(a)gmail.com> wrote: >> On 2013-11-08 18:38, Fred Lunnon wrote: >>> For arbitrary sphere radius r , the curve has four nodal singularities >>> where x = 0 or y = 0 : these are real and obvious where the curve >> intersects >>> itself when r > 1 , but safely complex and out of harm's way when r < 1 >> . >>> Rather unexpectedly, there are four more real singularities at the >>> inflections where z = 0 and x = y : these correspond to a pair of >>> lines >>> on the Enneper surface which superficially appear innocuous. >>> >>> There is a theorem that any TB curve must have (at least) four >> inflections, >>> see >>> http://www.qedcat.com/archive/165.html >>> --- it intrigues me that these points turn out to be so special here. >>> >>> Despite getting down and dirty with Magma's algebraic geometry feature, >>> I haven't yet managed to decide whether there are further singularities. >>> [Don't even think about trying to understand what schemes are --- just >> scope >>> the examples, then hack them!] >>> >>> Having gone this far, I couldn't resist putting the r ~ >> 0.2307718797455473 >>> curve onto a sphere. OK, I know it looks boring --- that's the whole >> point! >>> https://www.dropbox.com/s/ypsqc07tisw2gf2/tennis_ball.jpg >>> >>> Fred Lunnon >>> >> That's a pretty curve, but it accommodates a large inscribed circle, which >> requires large stretching to "map" onto the sphere. Minimizing stretching >> might motivate those racetrack shaped pieces we've seen. >> Recall years ago when the subject was the baseball stitch, we came up with >> *one piece* covers of arbitrarily low stretch, based on sphericons and >> fruit peels. >> gosper.org/onepiececover.png , gosper.org/tennis.gif >> It seems to me someone should manufacture tennis balls like this, just for >> the >> novelty. >> --rwg >>>

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