Is Simon's question this deep? Not simply analogous to, e.g., In[274]:= ContinuedFraction[(2 + Sqrt[5])^9] Out[274]= {439204, {439204}} ? --rwg On 2015-11-12 12:32, Dan Asimov wrote:
P.S. Whose continued fraction to 100 terms is:
[3; 3, 7, 4, 2, 30, 1, 8, 3, 1, 1, 1, 9, 2, 2, 1, 3, 22986, 2, 1,
32, 8, 2, 1, 8, 55, 1, 5, 2, 28, 1, 5, 1, 1501790, 1, 2, 1, 7, 6, 1,
1, 5, 2, 1, 6, 2, 2, 1, 2, 1, 1, 3, 1, 3, 1, 2, 4, 3, 1, 35657,
1, 17, 2, 15, 1, 1, 2, 1, 1, 5, 3, 2, 1, 1, 7, 2, 1, 7, 1, 3,
25, 49405, 1, 1, 3, 1, 1, 4, 1, 2, 15, 1, 2, 83, 1, 162, 2, 1, 1, 1, ...]
—Dan
On Nov 12, 2015, at 12:27 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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
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