No specific idea, but it's interesting to consider continued fractions as a dynamical system: F: (0,1) —> (0,1) via F(x) = 1/x - floor(1/x) Then as Gauss discovered and it's fun to prove, there is a measure M on (0,1) that is preserved by F. Namely, M([a,b)) := (1/ln(2)) Integral_{a <= x < b} dx/(1+x). The factor of 1/ln(2) is just to make the total measure come out to be 1. ( F preserves the measure M in the sense that M(Finv(([a,b)) = M([a,b)) for all intervals [a,b) \sub (0,1). ) (This is discussed in the incredibly enjoyable book by Mark Kac, "Statistical Independence in Probability, Analysis and Number Theory", 1959.) With respect to a preserved measure (having total measure = 1), a dynamical system can have a number of interesting properties: 1. Ergodicity. This means that any sets that are invariant under the dynamical system have measure 0 or 1. The dynamical system F: (0,1) —> (0,1) is in fact ergodic. IF a dynamical system 2. Mixing. This means that for any two measurable sets A, B of measures > 0, the intersection B \int [the image of A under the dynamical system] approaches having its measure = the product of [the measure of A]* [the measure of B], as time —> oo. This can be interpreted as saying that for any two events A and B of positive measure, the image of A under the dynamical system approaches being an independent event with respect to the event B, as time —> oo. I don't know if this holds for F: (0,1) —> (0,1). Does it? Remark: It's easy to see that mixing implies ergodicity, but not conversely. It's less easy to find an example of a dynamical system that is mixing (but it's a reasonable exercise). —Dan
On Nov 12, 2015, at 11:30 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote: ... here is something which is a puzzle to me,
I was exploring numbers like Fibonacci(k)/(Lucas(k-n)*((1+sqrt(5))/2)^n, where n is small and k >> 1.
Here is the odd thing : if you expand for example the number
5193981023518027157495786850488117/7177905237579946589743592924684178/(1/2+1/2*5^(1/2))^2 into a continued fraction, the surprise comes from the partial quotients of that expansion. It is quite chaotic. The maximal value being 83364870763649235403921261388869364666045817819140268784224747492762,
What is this ? how come a simple number like a/b*sqrt(5) has a c.frac expansion which such values ? I thought that approximations of a number like sqrt(5) could not be like that.
A quick examination shows that the size of these numbers (the maximal value of the c.f expansion) will be like Fibonacci(k)^2 (if n is small). In this example we have,
n = 163 and n = 2.
Can someone tell me how is this possible ?
I really don't see a general formula, since for some values of n and k, the behavior of the c.f. is quite <normal> with no high values, what are the conditions to have the maximal value ? I made some programs to analyze this and found only bizarre examples.
Best regards,
ps : I am back on the math-fun list after a quick absence.