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!! Watch the recurrence slam into reverse at n=57! In[17]:= Reap[Nest[{#[[1]] + 1, Sow[N[Numerator[#[[2]]]]]; (5 \[Sqrt](-100 + 9 #[[2]]^2 + 4 #[[2]]^4 - 12 (-1)^( 1/8 (-2 + #[[1]]) (1 + #[[1]]) (3 + #[[1]]) (4 + #[[ 1]])) #[[2]] Sqrt[-25 + #[[2]]^4]))/(Abs[-5 + 2 #[[2]]] (5 + 2 #[[2]]))} &, {3, 41/12}, 57]][[2,1]] // tim During evaluation of In[17]:= 0.042828,57 Out[17]= {41., 11285., 3.34416*10^6, 4.45722*10^10, 6.54686*10^14, 3.12099*10^20, 2.49851*10^26, 5.00963*10^33, 1.60444*10^41, 1.784*10^50, 2.09239*10^59, 1.09929*10^70, 6.53153*10^80, 1.21849*10^93, 3.89694*10^105, 3.13867*10^119, 3.92379*10^133, 1.76084*10^149, 8.19116*10^164, 1.69312*10^182, 4.07077*10^199, 2.97577*10^218, 3.79435*10^237, 1.22952*10^258, 5.99781*10^278, 1.08505*10^301, 2.006209100334819*10^323, 1.628644108339963*10^347, 1.584775641701988*10^371, 4.546385929353489*10^396, 2.306385900107065*10^422, 3.011027064913547*10^449, 5.731163695214867*10^476, 4.174132093433851*10^505, 3.074040656862112*10^534, 9.78526773250994*10^564, 3.853394734453615*10^595, 4.345605709540367*10^627, 8.752339001418554*10^659, 4.609196830221771*10^693, 3.423848362005287*10^727, 1.002440705228261*10^763, 2.946534125540071*10^798, 3.672615511787510*10^835, 5.851420618448239*10^872, 2.598805008061456*10^911, 2.073644300832899*10^950, 4.409690172448374*10^990, 1.278984026197366*10^1031, 1.502854321987985*10^1073, 1.766611469306492*10^1115, 8.611636998530298*10^1158, 5.548598132260577*10^1202, 9.72410563316432*10^1247, 5.548598132260577*10^1202, 8.611636998530298*10^1158, 1.766611469306492*10^1115} The usual penalty for choosing the wrong sign is surds. Before focusing on 57, I was running out to 9999 trying to figure out why the numerators weren't growing faster. How did it never surd? At any rate, the formula can be (grotesquely} rescued by trying both signs and choosing the rational with larger numerator. --rwg On Mon, Jun 30, 2014 at 4:06 AM, Bill Gosper <billgosper@gmail.com> wrote:
In what was probably Europe's first public math contest, the great Fibonacci was asked for three rational squares with common difference 5. (This is equivalent to finding a rational Pythagorean triangle with area 5.) Somehow, Fibonacci found (31/12)^2, (41/12)^2, (49/12)^2. This is the first of an infinite sequence of such solutions, found these days by repeated "addition" of the point {5*5/4, 6*5^2/4^2} along the "elliptic" curve y^2 = (x-5) x (x+5), producing the sequence 5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, ..., whose elements *of even index* are Fibonacci's desire. Elliptic curve addition doesn't quite provide a nice recurrence formula. Empirically, there's a mysterious, period 8 sign pattern: pisano[1] = 5/2; pisano[2] = 41/12; pisano[n_Integer] := (5 \[Sqrt](-100 + 9 pisano[-1 + n]^2 + 4 pisano[-1 + n]^4 - 12 (-1)^(1/8 (-2 + n) (1 + n) (3 + n) (4 + n)) Sqrt[-25 pisano[-1 + n]^2 + pisano[-1 + n]^6]))/ (Abs[-5 + 2 pisano[-1 + n]] (5 + 2 pisano[-1 + n]))
Try a few: Table[pisano[n], {n, 8}]
{5/2, 41/12, 11285/1562, 3344161/1494696, 44572169525/7118599318, 654686219104361/178761481355556, 312098738002194296165/128615821825334210638, 249850594047271558364480641/5354229862821602092291248}
Test the "even" ones: Sqrt[#^2 + {-5, 5}] & /@ %[[Range[2, 8, 2]]]
{{31/12, 49/12}, {113279/1494696, 4728001/1494696}, {518493692732129/178761481355556, 767067390499249/178761481355556}, {249563579992463717493803519/5354229862821602092291248, 250137278774864229623059201/5354229862821602092291248}}
Rich suggested the solutions might obey a Somos recurrence. So far, I've only found one (Somos5) for the numerators: a[n] -> (124558 a[-3 + n] a[-2 + n] - 781 a[-4 + n] a[-1 + n])/a[-5 + n] The 124558 and 781 are very probably minimal. "Fitting" a 𝝑 fcn to these numerators held a surprise: a[0], a[1], a[2], ... , a[n] = 1,5,41,11285,... =
(I (-1)^n 5^(1/2 Mod[n, 2]) E^(- I ArcTan[3/5]/2) (π/2)^(3/4) u^n^2* EllipticTheta[2, n z, (-1)^(-(13/34) + I/34)])/(2 17^(1/4) Gamma[5/4])
(Note the crazy (constructible!) 34th root of unity. Unfortunately, u and z are simply what they need to be to make a[1] and a[2] work. ISC doesn't recognize them, and I can't find a way to symbolically separate them. z -> 4.908105290682877137963508159098016296309287655031188041917052851155165374957836660235707132115102`69. -
0.197404925101320072627868756915313864010305639948302119404764160895568125915396715987289924743617`69. I, u -> -0.910971694653082823579468669395203737157839231764166792826687755862129892782777001356185727331806`65.8+
1.430029427800740885258427207242669226859970344275381501850156729643487583694210398313066894949425`66. I}
Mathematica's Root notation lets you exactly specify an implicitly defined root, say, the solution to a transcendental equation. Newton's method is getting to be like another button on your calculator. But how do we handle the solution vectors of simultaneous transcendental (say) equations, like for u and z here? --rwg (with much help from NeilB)