# [seqfan] Re: Aperiodic k-Simple Continued Fractions?

Joerg Arndt arndt at jjj.de
Sun May 30 10:21:45 CEST 2010

```This might be of interest:
http://demonstrations.wolfram.com/InstantlyPeriodicGeneralContinuedFractionRepresentationsOfSq/

" This Demonstration provides a way to instantly write a periodic
general continued fraction representation of square roots. This can be
used as a quick way to easily approximate the numerical value of
square roots.  (As opposed to a general continued fraction, a simple
continued fraction has all of its numerators equal to 1.)
[snip expression as images] "

* Paul D Hanna <pauldhanna at juno.com> [May 04. 2010 09:44]:
> 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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