November 2016 - mathfun@mailman.xmission.com

[math-fun] The Hippasus Integers
by Ed Pegg Jr 18 Nov '16

18 Nov '16

The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers. It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ... The ℚ(√−2) numbers are now known as Hippasus integers. Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use. I calculated the nine types of Heegner primes and plotted them out. http://community.wolfram.com/groups/-/m/t/965609 Anyone agree / disagree about calling ℚ(√−2) the Hippasus integers? Ed Pegg Jr

3 2

[math-fun] The Hippasus Integers
by Bill Gosper 18 Nov '16

18 Nov '16

Date: 2016-11-17 18:43 From: Ed Pegg Jr <ed(a)mathpuzzle.com> The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers. It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ... The ℚ(√−2) numbers are now known as Hippasus integers. Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use. I calculated the nine types of Heegner primes and plotted them out. http://community.wolfram.com/groups/-/m/t/965609 Anyone agree / disagree about calling ℚ(√−2) the Hippasus integers? Ed Pegg Jr --------------- His assailants probably didn't kill him for proving √2 irrational, but rather for taunting them for being unable to state the numerator and denominator. --rwg

1 0

[math-fun] Hyperbolic paraboloid
by James Propp 18 Nov '16

18 Nov '16

John Partridge asked me about the hyperbolic paraboloid shown in http://pyrigan.com/wp-content/uploads/2016/10/CelebrationOfMind.pdf Is the 2D projection of the hole a true circle? See page 12. Jim Propp

5 5

[math-fun] Mapping problem
by David Wilson 17 Nov '16

17 Nov '16

Let S(n) be the largest subset of Z(n) fixed by the mapping n -> n^2, and let f(n) = |Z(n)|. For example, S(25) = {0, 1, 6, 11, 16, 21} is the largest set of residues modulo 25 fixed by the mapping n -> n^2, so f(25) = |S(25)| = 6. Can you find a formula for f(n) in terms of n?

4 5

[math-fun] Landau's function of 2^16
by Allan Wechsler 17 Nov '16

17 Nov '16

I got an email from Claus Michael Ringel at the University of Bielefeld, asking me if I knew a reasonable way to evaluate Landau's function (A000793) at 2^16 = 65536. This is the maximum least common multiple of any partition of 65536; equivalently it is the maximum order of any element of Sym(65536). I am sure there are lots of shortcuts, that save one from enumerating all the partitions of 65536. Can anyone suggest one? (Of all the code snippets in the relevant page at A000793, all of them except one are clearly just enumerating partitions and comparing LCM's. But Charles Greathouse's PARI code is not doing anything like that, and I don't have a clue what it *is* doing.)

6 22

[math-fun] Not particularly math, possibly fun, particularly particles
by Bill Gosper 16 Nov '16

16 Nov '16

Right. Thanks--that explains everything. But it would be so cool to have exact reversibility. To (perhaps unnecessarily) repeat: Might it be possible to update the particles sequentially a la the (two skew) Minsky "circle" algorithm: newx -> x - floor(d y), newy -> y + floor(e newx) which is exactly reversible, no matter how batsh*t the floor function is. The penalty for this is (very remediable) non-circularity. Could we update the particles with sequential (and thus reversible) skews? If so, there are probably remedies for any visible imperfections in the forward simulation. --rwg On 2016-11-15 11:06, Mike Speciner wrote: > Bill, I'm assuming you created this by time-reversing an initial > configuration. If so, .... > The three balls that end up apparently motionless and linearly > together at the lower left aren't really quite as motionless or > linearly together as they appear, presumably due to tiny floating > point "errors" that accumulate in both the reverse and the forward > simulation. (You can examine the actual positions and velocities by > mousing over the balls with the simulation paused.) In the forward > simulation, there are probably no simultaneous collisions. In any > case, only one collision (the earliest) is processed at a time, which > might occur at a small fraction of the time step. Then all the > positions and velocities are updated to the collision time post > collision, and the process is repeated until all the collisions within > the stepsize have been processed. > > --ms > > On 14-Nov-16 19:39, Bill Gosper wrote: >> Simple experiment; negative result: I can't see a difference between >> >> title:gas2 >> clearwalls >> clearballs >> time:0 >> speed:4 >> wall:{x:[[-256,-256],[-256,256]]} >> wall:{x:[[-256,256],[256,256]]} >> wall:{x:[[256,256],[256,-256]]} >> wall:{x:[[256,-256],[-256,-256]]} >> wall:{x:[[0,56],[-56,0]]} >> ball:{x:[138.95544361180137,-53.80397594446178],v:[1.510335795838575,25.481978458903548],m:1,r:10,c:"gray"} >> ball:{x:[-134.9335319061281,-186.86010684600106],v:[21.85076947267741,9.512509711816417],m:1,r:10,c:"gray"} >> ball:{x:[54.22470051433805,245.51158342487696],v:[-24.566342384761167,-19.008502444414795],m:1,r:10,c:"gray"} >> ball:{x:[-10.77118714505185,226.1293232034858],v:[-22.55619118918868,-2.3002862909202832],m:1,r:10,c:"gray"} >> ball:{x:[-75.57273059907199,-225.01762417294833],v:[-1.2330104812166782,-0.14817511072452252],m:1,r:10,c:"gray"} >> >> and the same text with the last five lines reversed. The continuation from >> time 400 should have one ball shuttling >> across the bottom for many periods. Do you presort the balls? Take pains >> to simulate parallel update? Either >> would foil my plan. >> --rwg >> One non-surprise: speed -2 is a poor slo-mo time-reversal of speed 4. >> >> >> >> On Sun, Nov 13, 2016 at 10:01 PM, Bill Gosper <billgosper(a)gmail.com> wrote: >> >>> Maybe we can at least get Minsky-style reversibility by updating the >>> particles in reverse order if >>> time runs negative. --rwg >>> I somehow missed the Load from file button! My file is small enough to >>> fit in a url. >>> Is there a way for it to override the default initial configuration? >>> >>> Date: 2016-11-12 13:44 >>> From: Mike Speciner <ms(a)alum.mit.edu> >>> To: math-fun <math-fun(a)mailman.xmission.com> >>> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >>> >>> If you SHOW AS TEXT, you can cut and paste to save in a file of your >>> choosing (and edit it as you choose). You can load such a file with the >>> "Load from file" function. (The LOAD and SAVE functions use cookies, with a >>> nonstandard encoding to reduce their size.) >>> >>> Good question about perfect computation with rationals. There are an awful >>> lot of square roots in the calculations; I'm not sure the conditions under >>> which they don't produce irrationals. I suspect one dimensional scenarios >>> are more likely to be able to produce exact results. (It's been a long time >>> since I looked at this code. I do remember deliberately not adding in >>> gravity because that made reversibility harder. The PDP-6 version had >>> gravity. And I vaguely remember having some code to make sure energy was >>> [nearly] conserved, but I'm not sure it's in this new version.) >>> >>> --ms >>> >>> On 12-Nov-16 01:44, Bill Gosper wrote: >>>>> Date: 2016-11-05 09:51 >>>>> From: Mike Speciner <ms(a)alum.mit.edu> >>>>> https://ms0.github.io/gas.html >>>> This is a great toy! I delete everything except six identical balls (and >>>> maybe one wall), set one ball >>>> to bounce around for a while before disrupting the others, run it half a >>>> minute into the disruption, save >>>> it out, and show it time-reversed to some unsuspecting person. Or if I'm >>>> afraid they'll notice the >>>> negative time passage, I manually negate all the velocities before >>> saving. >>>> (Is there a way to load >>>> one of those text files?) >>>> >>>> Unfortunately, neither of these reversal stunts is perfect, due to >>>> accumulated numerical errors, >>>> which I assume you've worked hard to minimize. >>>> >>>> So let's get particularly math-y and ask: Given rational positions and >>>> velocities, can we simulate >>>> one of these six-ball scenarios exactly and indefinitely, without runaway >>>> denominator buildup, >>>> nor irrational collision results? --rwg >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

2 1

Re: [math-fun] Mapping problem
by Dan Asimov 16 Nov '16

16 Nov '16

I'm not feeling particularly patient at the moment. Can you kindly say what the notation "Z(n)" stands for? I can try to guess, of course, but I would rather not. —Dan ----- Let S(n) be the largest subset of Z(n) fixed by the mapping n -> n^2, and let f(n) = |Z(n)|. For example, S(25) = {0, 1, 6, 11, 16, 21} is the largest set of residues modulo 25 fixed by the mapping n -> n^2, so f(25) = |S(25)| = 6. Can you find a formula for f(n) in terms of n? -----

5 6

[math-fun] "Pushing" and "popping" for deductions about inter-referential sentences
by James Propp 16 Nov '16

16 Nov '16

I just realized (painfully close to my deadline) that my essay on self-referential sentences could really benefit from a terminological distinction that I may have to coin. It's the distinction between Deduction style #1: If P says "Q is true", and P is true, then Q is true. and Deduction style #2: If P says "Q is true", and Q is true, then P is true. I'm tempted to call them "pushing" and "popping", respectively. And now that I think about it, I'm wondering whether Hofstadter used exactly this nomenclature in "Godel, Escher, Bach". I can't decide whether it's merely the sort of thing Hofstadter would do, or something he actually did! And I can't seem to locate my copy of GEB. Can someone with a handy copy look up "push" and "pop" in the index? Thanks, Jim Propp

3 3

[math-fun] Not particularly math, possibly fun, particularly particles
by Bill Gosper 15 Nov '16

15 Nov '16

Maybe we can at least get Minsky-style reversibility by updating the particles in reverse order if time runs negative. --rwg I somehow missed the Load from file button! My file is small enough to fit in a url. Is there a way for it to override the default initial configuration? Date: 2016-11-12 13:44 From: Mike Speciner <ms(a)alum.mit.edu> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> If you SHOW AS TEXT, you can cut and paste to save in a file of your choosing (and edit it as you choose). You can load such a file with the "Load from file" function. (The LOAD and SAVE functions use cookies, with a nonstandard encoding to reduce their size.) Good question about perfect computation with rationals. There are an awful lot of square roots in the calculations; I'm not sure the conditions under which they don't produce irrationals. I suspect one dimensional scenarios are more likely to be able to produce exact results. (It's been a long time since I looked at this code. I do remember deliberately not adding in gravity because that made reversibility harder. The PDP-6 version had gravity. And I vaguely remember having some code to make sure energy was [nearly] conserved, but I'm not sure it's in this new version.) --ms On 12-Nov-16 01:44, Bill Gosper wrote: >> Date: 2016-11-05 09:51 >> From: Mike Speciner <ms(a)alum.mit.edu> >> https://ms0.github.io/gas.html > This is a great toy! I delete everything except six identical balls (and > maybe one wall), set one ball > to bounce around for a while before disrupting the others, run it half a > minute into the disruption, save > it out, and show it time-reversed to some unsuspecting person. Or if I'm > afraid they'll notice the > negative time passage, I manually negate all the velocities before saving. > (Is there a way to load > one of those text files?) > > Unfortunately, neither of these reversal stunts is perfect, due to > accumulated numerical errors, > which I assume you've worked hard to minimize. > > So let's get particularly math-y and ask: Given rational positions and > velocities, can we simulate > one of these six-ball scenarios exactly and indefinitely, without runaway > denominator buildup, > nor irrational collision results? --rwg

2 3

Re: [math-fun] Landau's function of 2^16,
by Tomas Rokicki 14 Nov '16

14 Nov '16

Sorry for the brevity I'm on a phone. This code computes g(n,p) which is the best g(n) where the factors are prime powers with the prime le p. It builds on this prime by prime. Then g(n,n) is g(n). On Nov 13, 2016 2:19 PM, "Tomas Rokicki" <rokicki(a)gmail.com> wrote: Read the code carefully. On Nov 13, 2016 2:17 PM, "Allan Wechsler" <acwacw(a)gmail.com> wrote: > Ooh, even worse: suppose N = p^k + M, and g(M) happens to be divisible by > p^k? Then the product p^k*g(M) would look like a really good candidate for > g(N), but would in fact have a spurious factor of p^k. > > On Sun, Nov 13, 2016 at 5:15 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: > > > It looks like this code depends on a lemma whose truth is not obvious to > > me. > > > > "For any N, there exists a prime power p^k such that g(N) = g(N-p^k) * > > p^k." > > > > It is conceivable to me that for any component p^k of g(N), g(N-p^k) > might > > not be relatively prime to p^k. That is, it's conceivable that none of > the > > cofactors of prime powers in g(N) are themselves optimal in the Landau > > sense. I admit that it doesn't seem _likely_, because you would have to > > have bad luck for every component p^k in g(N). But still conceivable. Am > I > > missing an easy demonstration of this lemma? > > > > On Sun, Nov 13, 2016 at 12:34 PM, Tomas Rokicki <rokicki(a)gmail.com> > wrote: > > > >> Perhaps this code will be simple enough for most to follow. > >> > >> Note that I use doubles and don't report the actual partition. > >> This is trivial to do (just add an array storing the last prime > >> power that improved a result, and then walk backwards from > >> there). > >> > >> This code easily handles 65536 and gives 2.41074e+388. > >> Running it in Python (with its superior number stack) should > >> give you an exact result pretty quickly; I can provide that if > >> anyone is interested. > >> > >> for (int i=0; i<N; i++) > >> maxv[i] = 1 ; > >> for (int i=2; i<=N; i++) { > >> if (!isprime(i)) > >> continue ; > >> for (int j=N-1; j>=i; j--) { > >> long double hi = maxv[j] ; > >> for (long long pp=i; pp<=j; pp=pp*i) > >> hi = max(hi, maxv[j-pp]*pp) ; > >> maxv[j] = hi ; > >> } > >> } > >> for (int i=0; i<N; i++) > >> cout << i << " " << maxv[i] << endl ; > >> > >> > >> On Sun, Nov 13, 2016 at 9:09 AM, Tomas Rokicki <rokicki(a)gmail.com> > wrote: > >> > >> > This is a pretty common programming contest problem. > >> > You can make an additional restriction as follows: > >> > > >> > Every factor (except perhaps the last) should be a power of a > >> prime. > >> > (If a factor is composite, you can factor it into prime powers > and > >> > get equivalent impact with fewer base elements.) > >> > > >> > Then, you can solve this problem using dynamic programming. > >> > First you solve it for n=1..65536 using only powers of 2. Then based > >> > on that you solve it for n=1..65536 using powers of 2 and powers of > >> > 3. Then you extend that with powers of 5, and so on and so forth; > >> > pretty straightforward. > >> > > >> > You do need to use a programming language or library that supports > >> > bigints, or else you can represent the product by its logarithm. > >> > > >> > This is not considered a difficult problem for modern college > >> > programming contests. > >> > > >> > On Sun, Nov 13, 2016 at 4:02 AM, Fred Lunnon <fred.lunnon(a)gmail.com> > >> > wrote: > >> > > >> >> I'm suffering some disconnection here: compare JA below > >> >> > >> >> >> The upper bound used for the downward search is truly enormous for > >> n = > >> >> 2^16. > >> >> > >> >> with https://oeis.org/draft/A000793 comments > >> >> > >> >> >> Grantham mentions that he computed a(n) for n <= 500000. > >> >> > >> >> --- citing a 1995 reference given later. > >> >> > >> >> Yet 2^16 = 65536 < 500000 , and JG's hardware 20 years older! > >> >> > >> >> WFL > >> >> > >> >> > >> >> > >> >> On 11/13/16, Joerg Arndt <arndt(a)jjj.de> wrote: > >> >> > * Allan Wechsler <acwacw(a)gmail.com> [Nov 13. 2016 08:33]: > >> >> >> [...] > >> >> >> > >> >> >> (Of all the code snippets in the relevant page at A000793, all of > >> them > >> >> >> except one are clearly just enumerating partitions and comparing > >> LCM's. > >> >> >> But > >> >> >> Charles Greathouse's PARI code is not doing anything like that, > and > >> I > >> >> >> don't > >> >> >> have a clue what it *is* doing.) > >> >> > > >> >> > Charles' code seems to use what I just entered as a formula in > >> A000793: > >> >> > For n>=2, A000793(n) = max_{k} A008475(k) <= n. > >> >> > Cf. https://oeis.org/draft/A000793 > >> >> > > >> >> > Is this correct? > >> >> > > >> >> > The upper bound used for the downward search is truly enormous for > n > >> = > >> >> > 2^16. > >> >> > > >> >> > > >> >> >> _______________________________________________ > >> >> >> math-fun mailing list > >> >> >> math-fun(a)mailman.xmission.com > >> >> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> >> > > >> >> > _______________________________________________ > >> >> > math-fun mailing list > >> >> > math-fun(a)mailman.xmission.com > >> >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> >> > > >> >> > >> >> _______________________________________________ > >> >> math-fun mailing list > >> >> math-fun(a)mailman.xmission.com > >> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> >> > >> > > >> > > >> > > >> > -- > >> > -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link > >> > suppressed; ask me why] -- > >> > > >> > >> > >> > >> -- > >> -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link > >> suppressed; > >> ask me why] -- > >> _______________________________________________ > >> math-fun mailing list > >> math-fun(a)mailman.xmission.com > >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >> > > > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

2 1