[seqfan] Aperiodic k-Simple Continued Fractions?

Paul D Hanna pauldhanna at juno.com
Tue May 4 05:34:05 CEST 2010


SeqFans, 
   It is intuitive that the "simple" continued fraction expansion 
of a quadratic irrational should be periodic. 
After all, x = 1/(a + 1/(b + x)) = (sqrt(b^2 + 4b/a) - b)/2 
is one example that obviously leads to a periodic simple CF. 
  
Consider the continued fraction expansion described by: 
   x = c + k/(q1 + k/(q2 + k/(q3 + k/(q4 + ... + k/(q_n + ...))))) 
where k is a fixed positive integer and x is an irrational constant; 
call this the "k-simple" continued fraction of x, denoted by:
   x = [k| c; q1, q2, q3, q4, ...], 
where {q_n >= k} are the k-simple partial quotients. 
  
Surprisingly, it appears that the k-simple CF of sqrt(n) is aperiodic 
in some cases; below are a few examples. 
 
I do not know for sure that they are indeed aperiodic; 
could they actually be periodic after some point? 
  
Please direct me to a source if this is well-known 
in some obscure hall of higher learning. 
   Paul 
 
EXAMPLES. 
  
EX.1. 
The 2-simple continued fraction of sqrt(92) begins:
sqrt(92) = [2| 9;3,5,7,2,2,3,6,2,5,2,7,8,2,41,4,3,2,15,11,
2,31,5,17,2,4,2,3,2,3,2,6,3,6,4,63,454,12,3,6,
4,2,5,8,2,4,4,55,17,6,3,10,2,16,2,3,2,3,2,3,
15,4,2,3,2,7,7,199,4,2,11,3,2,2,2,2,5,4,7,5,
7,10,24,10,258,4,3,14,2,3,17,25,3,19,2,2,3,3,26,3,
2,4,2,3,7,4,2,26,3,2,3,5,4,3,31,10,10,6,3,34,
7,4,2,3,10,15,5,2,18,6,2,2,8,10,2,4,2,2,3,10,
2,8,10,2,16,3,61,2,4,2,2,6,2,3,40,3,3,4,2,3,
2,27,2,2,22,5,5,2,5,4,2,9,9,7,2,7,5,4,4,2,
2,8,2,8,8,4,2,10,8,15,6,5,21,2,15,2,4,2,12,4,...].
 
EX.2. 
The 3-simple continued fraction of sqrt(21) begins:
sqrt(21) = [3| 4;5,20,49,8,4,6,5,5,26,3,7,9,5,3,14,4,4,14,3,
184,4,5,7,3,5,3,3,9,9,5,5,11,3,4,4,69,25,30,5,
3,10,14,4,27,3,4,3,6,14,31,25,12,4,4,6,16,6,15,6,
21,5,360,11,11,11,6,5,4,4,3,33,76,4,41,3,3,23,3,7,
6,6,29,3,28,11,3,3,6,3,8,7,4,3,12,33,3,26,4,3,
68,8,4,3,34,38,4,37,11,11,3,12,3,56,3,3,4,14,21,4,
4,8,3,25,12,9,10,4,14,6,3,3,7,239,4,42,3,22,13,15,
5,10,7,44,5,3,3,9,65,5,3,11,8,3,4,4,3,3,9,12,
83,9,12,3,3,50,13,23,4,173,96,4,4,3,3,7,3,15,9,4,
3,7,5,4,7,3,6,4,4,75,4,50,4,3,10,3,9,6,4,23,...].
 
EX.3. 
The 5-simple continued fraction of sqrt(5) begins:
sqrt(5) = [5| 2;21,27,6,5,8,16,10,11,13,5,22,9,7,7,8,25,5,5,7,
17,9,6,5,52,5,5,10,5,17,33,178,17,21,6,6,5,7,8,65,
5,5,5,26,7,7,8,9,9,10,5,7,144,24,6,22,41,11,5,5,
6,6,11,5,5,6,37,5,9,6,7,59,6,9,32,5,47,6,5,6,
6,32,5,8,16,65,5,6,5,14,6,24,24,7,8,7,5,10,7,6,
8,30,7,10,6,14,240,14,11,66,13,38,13,5,15,5,158,89,5,6,
45,21,8,13,15,126,5,6,8,9,5,9,23,10,5,44,6,19,23,11,
7,35,26,10,83,30,10,9,19,5,7,16,25,14,17,8,46,51,54,6,
7,8,9,36,6,12,5,34,17,12,5,16,11,5,9,9,35,6,12,9,
11,19,84,6,5,34,40,8,45,61,5,26,7,82,12,7,9,10,10,16,...].
 
EX.4. 
The 8-simple continued fraction of sqrt(2) begins:
sqrt(2) = [8| 1;19,25,15,8,21,11,12,8,83,14,27,44,39,12,43,18,9,8,80,
9,12,17,53,8,39,8,53,37,79,399,12,23,35,14,17,57,8,26,29,
19,9,12,18,14,81,11,8,22,8,15,12,27,568,19,12,15,17,14,8,
26,227,65,8,27,10,25,11,14,16,55,10,11,23,21,10,64,10,2024,12,
10,22,9,36,101,29,12,37,34,13,14,9,174,9,10,25,8,26,648,15,
9,23,9,12,105,30,14,8,28,27,18,8,17,53,13,14,23,10,18,12,
199,9,30,12,21,13,29,8,8,17,79,29,23,8,36,23,9,2427,26,30,
20,10,22,32,21,12,13,42,45,74,26,11,18,20,291,27,34,8,10,15,
35,11,13,8,11,26,49,14,20,11,19,9,18,14,9,13,55,14,29,8,
19,189,14,2147,9,106,10,18,9,43,10,396,14,29,57,148,14,8,27,14,...].
  
[END]


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