June 2011 - mathfun@mailman.xmission.com

Re: [math-fun] Favourite prime numbers
by Bill Gosper 03 Jun '11

03 Jun '11

Mike Stay>My favorite prime is 1+i. One good reason: It's the base of a binary number system, with bits 0 and 1, that can represent all complex numbers (without needing a sign bit). The pure fractions represent a simply connected region coincident with the twindragon fractal (File:Dragon_tiling5.svg <http://en.wikipedia.org/wiki/File:Dragon_tiling5.svg>) . --rwg Start with 0. Then iterate Union[%, 1 + %]/(1 + I); a few times. Then ListPlot[{Re[#], Im[#]} & /@ %, PlotStyle -> PointSize[Large]] On Fri, Jun 3, 2011 at 3:01 PM, <rcs(a)xmission.com <http://gosper.org/webmail/src/compose.php?send_to=rcs%40xmission.com>> wrote:> Let's not forget 314159! --Rich>> -----> Quoting Thane Plambeck <tplambeck(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=tplambeck%40gmail.com>>:>>> 149459, a prime number, the period of the octal game 0.16>>>> It magically appeared in the green phosphor of a DEC VT100 in the>> Stanford CS department basement in the fall of 1986>>>>>>>> On Fri, Jun 3, 2011 at 6:26 AM, Victor Miller <victorsmiller(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=victorsmiller%40gmail.com>>>> wrote:>>>>>> My favorite prime is 144169. Besides the fact that it looks like 12^2>>> concatenated with 13^2, it's also the discriminant of the quadratic field>>> which contains the eigenvalues of the Hecke operators on modular forms of>>> weight 24.>>>>>> Victor>>>>>> On Fri, Jun 3, 2011 at 8:49 AM, Dan Asimov <dasimov(a)earthlink.net <http://gosper.org/webmail/src/compose.php?send_to=dasimov%40earthlink.net>> wrote:>>>>>>> (((>>>> My favorite number is 24 hands down, because>>>> of its amazing properties like showing up>>>> in Dedekind eta, 1^2 + ... + 24^2 = 70^2>>>> being the only such relation, the 24-dimensional>>>> Leech lattice, and 24 being the largest number N>>>> such that all smaller numbers K with GCD(K,N) = 1>>>> satisfy K^2 == 1 (mod N). At least some of these>>>> are interconnected.>>>> )))>>>>>>>> But how about prime numbers? After the first few,>>>> how do you distinguish, say, 101, 103, 107, 109 ?>>>> Are there standard measures of some sort that>>>> distinguish among prime numbers?>>>>>>>> --Dan>>>>>>>> Sometimes the brain has a mind of its own.>>>>

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Re: [math-fun] surfaces of revolution & differential equations
by Bill Gosper 03 Jun '11

03 Jun '11

VE> On Jun 2, 2011, at 5:15 PM, Bill Gosper wrote: > >> Eh? 7 is as symmetrical as you can get. > I miscounted: 3 and 7 are the only solutions with all equal radii (I'm not > including 2 because the solution is a continuum). >> >>> Fourteen appears to be the first instance where all the radii are >>> distinct. >> >> In your aluminum puzzle, two disks are almost indistinguishable, but not >> >> switchable, due to the precise machining. But two others *are* >> >> switchable and apparently indistinguishable. Are you saying they are >> >> mathematically distinct?? By how much? Do you have their actual >> polynomials? >> >> --Bill > The two disks you refer to have very slightly different radius, about > 0.013%. That's > about 0.1 mil at the scale of the aluminum puzzle -- beyond what can be > machined > or discerned by hands and eyes. That's amazing! Especially since they seem to lie diametrically opposed. > I never worked out the polynomials. > > Veit All we need are a few hundred digits and FindIntegerNullVector . (And a bit of shamelessness.) --rwg

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Re: [math-fun] surfaces of revolution & differential equations
by Bill Gosper 02 Jun '11

02 Jun '11

Quoting Veit Elser <ve10(a)cornell.edu <http://gosper.org/webmail/src/compose.php?send_to=ve10%40cornell.edu>>: > Bill,>> The server is back up and yes, the solutions for 4, 5, 7, 8, ...> disks are examples of symmetry breaking (the radii are not equal> in the configurations that maximize the radius sum). Eh? 7 is as symmetrical as you can get. > Fourteen appears to be the first instance where all the radii are distinct. In your aluminum puzzle, two disks are almost indistinguishable, but not switchable, due to the precise machining. But two others *are* switchable and apparently indistinguishable. Are you saying they are mathematically distinct?? By how much? Do you have their actual polynomials? --Bill BtW, the mnemonic "1,4" encodes a seed configuration from which the entire solution follows almost immediately. >> Since fourteen is the first number where symmetry is completely> broken, that would make it my favorite number.>> My place of work was founded on a principle expressed in fourteen> words.>> Veit>> On Jun 1, 2011, at 5:30 AM, Bill Gosper wrote:>>> Veit>>>> Is your five disk packing of the unit disk maximizing sum(radii)>>>> (http://gosper.org/HTMLFiles/5disks.gif) an example of either of>>>> these asymmetries? How about your semisecret fourteen disk solution,>>>> which has no symmetry at all? (http://milou.msc.cornell.edu/images/>>>> seems to be down.)>>>> --rwg

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Re: [math-fun] surfaces of revolution & differential equations
by Bill Gosper 02 Jun '11

02 Jun '11

Veit>On May 31, 2011, at 11:35 AM, Allan Wechsler wrote: acw> Blah, I answered this just to Henry. I thought of the p-orbitals of> electrons, and some vibrational modes of a circular drum, and was sure there> were simpler examples of symmetry-breaking but couldn't construct one off> the top of my head. ve>Those are actually not examples of "symmetry-breaking" . The degeneracy (in energy/frequency) is just a manifestation of the symmetry, and indicates the system is especially sensitive to perturbations (that typically "break" the symmetry). Here's an example of true symmetry breaking: the ground state of the lithium trimer (three lithium atoms). If we neglect quantum motion of the nuclei, then the ground state shape is an isosceles triangle. Veit Is your five disk packing of the unit disk maximizing sum(radii) (http://gosper.org/HTMLFiles/5disks.gif) an example of either of these asymmetries? How about your semisecret fourteen disk solution, which has no symmetry at all? (http://milou.msc.cornell.edu/images/ seems to be down.) --rwg

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Re: [math-fun] Favourite number
by James Propp 02 Jun '11

02 Jun '11

Arg! I meant "Ying Tong Iddle I Po", but somehow typed "Ying Toddle Iddle I Po" (and then somehow failed to notice the typo). Jim

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Re: [math-fun] Favourite number
by James Propp 02 Jun '11

02 Jun '11

Lee Sallows writes: >Okay I'll have a look at your survey. Actually, my favourite number is >Beethoven's Moonlight Sonata, but I think Ying Tong Yiddle I Po comes in a >close second. I believe the correct lyrics are Ying Toddle Iddle I Po. (Though I suppose there could be a Hasidic version "Ying Tong Yidl I Po".) Jim Propp

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[math-fun] Favourite number
by Alex Bellos 02 Jun '11

02 Jun '11

I've set up an internet survey about favourite numbers: www.favouritenumber.net The idea emerged since I get asked a lot what my favourite number is. I dont have one, but I noticed that not only do most people expect a maths writer to have one, but they have deeply held convictions about their own favourite numbers. So, I thought I would try to quantify the world's favourite numbers. Ambitious I know, but in 24 hours since the site went live I have had about 3500 entries, mostly from the UK and USA, but also from dozens of other countries. Already interesting patterns are emerging, as well as interesting reasons for choosing the numbers. The survey i think will be interesting more for its qualitative results. If you dont have a favourite number, you are not excluded from taking part. There is a box for you. Please take part and share the link! I hope to have results ready for next year. Alex

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[math-fun] Collatz proof?
by Dan Asimov 01 Jun '11

01 Jun '11

Someone has posted a paper to the Web purporting to solve the Collatz problem, sometimes known as the 3x+1 conjecture: Abstract: Berg and Meinardus, 1994, 1995, introduced a pair of linear functional equations, acting on a space of holomorphic functions H, deﬁned on the open unit disk D. The simultaneous solutions of the two functional equations contain a simple two dimensional space, denoted by delta 2, and if one could show that delta 2 is the only solution, the Collatz conjecture would be true. Berg and Meinardus already presented the general solutions of the two individual functional equations. The present author reformulates the pair of functional equations in form of two linear operators, denoted by U and V . Thus, K := {h in H : U[h] = 0, V [h] = 0} is of main interest. Since the general solutions of U[h] = 0, V [h] = 0, denoted by KU , KV , respectively, are already known, we compute U[h] for h in KV and study the consequences of U[h] = 0. We show, that, indeed delta 2 = K follows, which implies that the Collatz conjecture is true." < http://preprint.math.uni-hamburg.de/public/papers/hbam/hbam2011-09.pdf > I haven't heard anything yet about expert opinion of its validity. --Dan Sometimes the brain has a mind of its own.

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