[math-fun] about some continued fractions related to sqrt(5).
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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.
The other thing that comes to mind is that the cf is pretty unremarkable except for the one ginormous term, and that term is pretty much the same length as the big rational factor: 5193981023518027157495786850488117/7177905237579946589743592924684178 83364870763649235403921261388869364666045817819140268784224747492762 I'd bet that most of the information in that term comes from the rational multiplier itself. 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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Hello, the actual number is : 10766857856369919884615389387026267/5193981023518027157495786850488117+3588952618789973294871796462342089/5193981023518027157495786850488117*5^(1/2) which is a/b + c*gr/d, a,b,c,d being integers, gr = golden ratio. of course, one thing I made was a program to check various exponents and values of n and k (previous expression), the maximum seems to be when n is small. ... but not all the time. now we expect the expression to have a big initial term, actually in that example, 3 is the first term, and then the term : 83364870763649235403921261388869364666045817819140268784224747492762 appears 3 times in the first 54293 terms of the continued fraction. We all know that quadratic irrationals have a periodic continued fraction, and it is easy to show that. What is surprising is the length of the period of that example and the height of the maximal term. example : sqrt(3442321), it has 3710 as maximal value and the period occurs 46 times over 100000 terms, I just took a random example. this is strange, the first term of the expression is 10766857856369919884615389387026267/5193981023518027157495786850488117 a rational which has a FINITE c.f. the second term : 3588952618789973294871796462342089/5193981023518027157495786850488117*5^(1/2) is periodic (in the c.f. ) and the maximum is 16672974152729847080784252277773872933209163563828053756844949498552. bonne journée, Simon Plouffe
participants (5)
-
Dan Asimov -
Dan Asimov -
Mike Stay -
rwg -
Simon Plouffe