Whoa, take my "aperiodic" radicands 19, 27, 31, 61, 71, 76, 85, 86, 87, ... with a grain of salt. I had a period cutoff of 1000, and √103 just came in with period 951 ! --rwg On 2016-02-04 19:03, Bill Gosper wrote:
Let x>=1. Iterate x <- 1/(x/2^n -1), where 2^n is the largest power of 2 <=x . The sequence of n's quantifies x analogously to a continued fraction. http://www.tweedledum.com/rwg/cfup.htm incorrectly stated that this sequence is periodic for quadratic surd x. The first x^2 = integer counterexamples appear to be 13, 14, 15, 29, 31, 35, 39, 41, ..., but they might just have huge periods. E.g., In[435]:= NestList[Simplify[1/(#/2^Floor[Log[2, #]] - 1)] &, Sqrt[167], 69]
Out[435]= {Sqrt[167], 8/(-8 + Sqrt[167]), 1/89 (39 + 8 Sqrt[167]), 1/46 (25 + 4 Sqrt[167]), 2/97 (21 + 4 Sqrt[167]), 1/79 (55 + 8 Sqrt[167]), 206 + 16 Sqrt[167], 128/(-25 + 8 Sqrt[167]), 1/(-1 + 128/(-25 + 8 Sqrt[167])), ( 2929 + 512 Sqrt[167])/5534, 5534/(-2605 + 512 Sqrt[167]), ( 8159 + 1024 Sqrt[167])/8119, (2 (8079 + 1024 Sqrt[167]))/13529, ( 2725 + 512 Sqrt[167])/5374, 5374/(-2649 + 512 Sqrt[167]), ( 8383 + 1024 Sqrt[167])/7663, (2 (6943 + 1024 Sqrt[167]))/16561, [...] 3363 + 1024 Sqrt[167])/13697, ( 5167 + 512 Sqrt[167])/2494, 9976/(-4809 + 512 Sqrt[167]), ( 6943 + 1024 Sqrt[167])/7663, (2 (8383 + 1024 Sqrt[167]))/13681, ( 2649 + 512 Sqrt[167])/5374, 5374/(-2725 + 512 Sqrt[167]), ( 8079 + 1024 Sqrt[167])/8119, (2 (8159 + 1024 Sqrt[167]))/13369, ( 2605 + 512 Sqrt[167])/5534, 5534/(-2929 + 512 Sqrt[167]), ( 6863 + 1024 Sqrt[167])/10063, 1/128 (25 + 8 Sqrt[167]), 128/(-103 + 8 Sqrt[167]), 2/97 (55 + 8 Sqrt[167]), 1/46 (21 + 4 Sqrt[167]), 2/89 (25 + 4 Sqrt[167]), 1/103 (39 + 8 Sqrt[167]), 1/8 (8 + Sqrt[167]), 16/(-8 + Sqrt[167]), 1/89 (39 + 8 Sqrt[167]), 1/46 (25 + 4 Sqrt[167])}
has menacing coefficients, but period 66. And, strangely, √13 +3, 1, -1, and -2 all have period 4. With large x, observable periods become increasingly sparse.
Negative n permit 0<x<1. Actually, NestList[Simplify[1/(1-#/2^Ceiling[Log[2, #]])] &, Sqrt[k], 69] is much richer in periodicity, the first apparent blowups being k = 19, 27, 31, 61, 71, 76, 85, 86, 87, ... . Near blowups make for longish periods, e.g. period(√65) = 127±, period(√69) = 395±, p(√70) = 399±, p(√75) = 310±. The patterns of rationals within the periods are intriguing. --rwg