Mike's suggestion may be prescient, as the real root of x^3 - 8x - 10 = 0 is K = (5-q)^(1/3) + (5+q)^(1/3) where q = sqrt(163/27) = sqrt(6 + 1/27) —Dan
On Nov 12, 2015, at 12:07 PM, Mike Stay <metaweta@gmail.com> wrote:
The 163 and large terms immediately makes me think of the "exotic" continued fraction for the real root of x^3 - 8x - 10 = 0.
On Thu, Nov 12, 2015 at 11:30 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello funsters,
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.
Simon Plouffe