On Fri, Jul 18, 2014 at 12:20 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Tue, Jul 1, 2014 at 5:14 PM, Bill Gosper <billgosper@gmail.com> wrote:
[Chop]
On Tue, Jul 1, 2014 at 1:49 AM, Bill Gosper <billgosper@gmail.com> wrote:
GAAAA--Rich privately suggested that the sign pattern in my 1st order, nonlinear recursion was not period 8, not even periodic. And he is right!!
How the freep did he know that? --rwg
The reason I got so exclamatory was the early (and middle) lack of supporting evidence or motivation for his claim. --rwg
Watch the recurrence slam into reverse at n=57!
[big chop]
NeilB seems to have it completely psyched. You just need to carry along your subscript (i.e., a counter): In[784]:= pisano[{n_, x_}] := {n + 1, Abs[(5 (3 x - psign[n]*2 Sqrt[x^4 - 25]))/(4 x^2 - 25)]}
[Terse, no?] In[787]:= NestList[pisano, {2, 41/12}, 4]
Out[787]= {{2, 41/12}, {3, 11285/1562}, {4, 3344161/1494696}, {5, 44572169525/7118599318}, {6, 654686219104361/178761481355556}}
This list actually starts {0,∞}, {1,5/2}, {2,41/12},..., if pisano bothered to take limits. In[790]:= Floor[Log[Numerator[#[[2]]]]] & /@ NestList[pisano, {2, 41/12}, 69]
Verify forward progress: Out[790]= {3, 9, 15, 24, 34, 47, 60, 77, 94, 115, 136, 161, 186, 214, 243, 275, 307, 343, 379, 419, 459, 503, 547, 594, 641, 693, 744, 799, 854, 913, 972, 1034, 1097, 1164, 1230, 1300, 1371, 1445, 1519, 1597, 1675, 1756, 1838, 1923, 2009, 2098, 2188, 2281, 2374, 2471, 2567, 2668, 2769, 2873, 2978, 3086, 3194, 3306, 3418, 3534, 3650, 3770, 3890, 4013, 4136, 4264, 4391, 4522, 4653, 4788}
This wouldn't be monotonic if psign ever lied. But wait, what's psign??
psign[n_] := Sign[Mod[n*-0.37271561316037777`, 1] - 1/2]
But what's -0.37271561316037777`??
In[775]:= N[InverseWeierstrassP[5^2/4*1/4, {5^2/4, 0}]/ (2 WeierstrassHalfPeriods[{5^2/4, 0}])][[1]] // Chop
Out[775]= -0.37271561316037793
which Neil got from staring at a very spongy Riemann surface. Note that this is "only" good out to about n=10^15. So why not leave it symbolic and let Mathematica gin up the requisite precision for the Mod and Sign? Notice the [[1]] near the end of %775. That quotient "N"s to an ordered pair instead of a number! And N seems to be the only access to it--you can't cleave the pair without committing to imprecision. And screwing around with Hold and ReleaseHold. --rwg
Neil's email (which he quietly sent while sitting right next to me) concludes: Additionally, given k=InverseWeierstrassP[<initial x>/4, {n^2/4,0}], we can compute the m'th elliptic curve x-coordinate with 4*WeierstrassP[m*k,{n^2/4,0}] and the m'th y-coordinate (y^2=x^3-n^2*x) with 4*WeierstrassPPrime[m*k,{n^2/4,0}] --Neil This completely finishes Fibonacci's problem of finding all triples of square rationals with common difference 5, the mth having √(central term) Out[811]= 2 Sqrt[WeierstrassP[2 m InverseWeierstrassP[{25/16, 75/32}, {25/4, 0}], {25/4, 0}]] The first eight are In[819]:= Table[Rationalize[%811, 69^-239], {m, 8}] Out[819]= {41/12, 3344161/1494696, 654686219104361/178761481355556, \ 249850594047271558364480641/5354229862821602092291248, \ 160443526614433014168714029147613242401001/\ 50016678000996026579336936742637753055940, \ 209239116668342644167838867143329714389679018137228536721441/\ 93092380947563478644577555596900542802151091304399908363272, \ 6531529540091581433784275365607334203207979539157560140956253170969369\ 04977985961/\ 1652129432363645568174549601966221192003955894694678397387353366810911\ 41974929836, 389694104145848748532083272246996368636625626448600416977271058482117\ 6712668535259971051251201565099266561/ 16701943947756425480481933369219751647526957097882440351972170228225\ 9070147808848039273138602363503527584} Test: In[820]:= Sqrt[#^2 + {5, -5}] & /@ % Out[820]= {{49/12, 31/12}, {4728001/1494696, 113279/ 1494696}, {767067390499249/178761481355556, 518493692732129/ 178761481355556}, {250137278774864229623059201/ 5354229862821602092291248, 249563579992463717493803519/ 5354229862821602092291248}, \ {195577262542844878506138849501555847171249/ 50016678000996026579336936742637753055940, 115038188620995226180802686473825513089249/ 50016678000996026579336936742637753055940}, \ {295147361324101461665473218814630755582253386512598845803201/ 93092380947563478644577555596900542802151091304399908363272, 21214405287844054428542609853501469645112322962237848990081/ 93092380947563478644577555596900542802151091304399908363272}, \ {750390141456429675692975482159216288564649702558340978265068525529492\ 577197267249/ 16521294323636455681745496019662211920039558946946783973873533668109\ 1141974929836, 53863920973762981989969801391321803641388652858263786597026709272685\ 4572430487071/ 16521294323636455681745496019662211920039558946946783973873533668109\ 1141974929836}, \ {391479590099159062446526431532215958267722907243296693773538849552903\ 3628971288541170127075160953800576001/ 1670194394775642548048193336921975164752695709788244035197217022822\ 59070147808848039273138602363503527584, 38790039977791415434943347292124944095416434812036821900993615287736\ 30968811107453857812241981709854955521/ 1670194394775642548048193336921975164752695709788244035197217022822\ 59070147808848039273138602363503527584}} I guess he emailed because he realized it would take a couple of days to sink in. --rwg All along, I was looking for a Somos-like solution with 𝝑 functions. So now my question is: Can ℘ express Somos?