Simon Plouffe: I was exploring numbers like Fibonacci(k)/(Lucas(k-n)*g^n) where n is small and k >> 1 and g=(1+sqrt(5))/2 is the golden ratio. WDS: Plouffe mistyped the above formula, as is evident from his mismatched parentheses. I have written what I am guessing he intended. Plouffe: The continued fractions of such numbers contain large partial quotients. WDS: Plouffe's example, rewritten, is 5193981023518027157495786850488117 / (3588952618789973294871796462342089* (3+sqrt(5))) = 0.27639... = [0; 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 83364870763649235403921261388869364666045817819140268784224747492762, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 4, 1, 2, 6, 2, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 13, 1, 7, 3, 1, 3, 7, 1, 12, 1, 1, 1, 5, 1, 2, 4, 2, 4, 2, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 4, 2, 4, 2, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 4, 2, 4, 2, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 4, 2, 4, 2, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 4, 2, 4, 2, 1, 5, 1, 1, 1, 11, 1, 1, 1, 5, 1, 2, 4, 6, 4, 7, 1, 6, 2, 2, 1, 31, 1, 2, 2, 6, 1, 7, 4, 6, 4, 7, 1, 6, 2, 2, 1, 31, 1, 2, 2, 6, 1, 7, 4, 6, 4, 7, 1, 6, 2, 2, 1, 31, 1, 2, 2, 6, 1, 7, 4, 6, 4, 7, 1, 6, 2, 2, 1, 31, 1, 2, 2, 6, 1, 5, 1, 2, 4, 3, 1, 3, 4, 2, 1, 5, 1, 4, 1, 2, 2, 2, 1, 1, 1, 29, 1, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 2, 4, 3, 1, 3, 4, 2, 1, 5, 1, 4, 1, 2, 2, 2, 1, 1, 1, 29, 1, 1, 1, 2, 2, 2, 1, 4, 1, 5, 1, 2, 4, 3, 1, 3, 4, 2, 1, 5, 1, 4, 1, 2, 2, 2, 1, 1, 1, 29, 1, 1, 1, 2, 2, 2, 1, 13, 1, 5, 1, 7, 4, 4, 2, 4, 4, 7, 1, 5, 1, 13, 1, 2, 2, 7, 1, 1, 1, 78, 1, 1, 1, 7, 2, 2, 1, 13, 1, 5, 1, 7, 4, 4, 2, 4, 4, 7, 1, 5, 1, 13, 1, 2, 2, 7, 1, 1, 1, 78, 1, 1, 1, 7, 2, 2, 1, 13, 1, 5, 1, 7, 4, 4, 6, 4, 4, 2, 1, 5, 1, 5, 1, 1, 1, 12, 1, 2, 6, 7, 1, 1, 1, 1, 1, 76, 1, 1, 1, 1, 1, 7, 6, 2, 1, 12, 1, 1, 1, 5, 1, 5, 1, 2, 4, 4, 6, 4, 4, 2, 1, 5, 1, 5, 1, 1, 1, 12, 1, 2, 6, 7, 1, 1, 1, 1, 1, 76, 1, 1, 1, 1, 1, 7, 6, 2, 1, 12, ... eventually periodic] The Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765,... each is sum of 2 preceding. The Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349,... each is sum of 2 preceding. So what is going on here? First of all, every number of Plouffe's class is a quadratic irrational, hence has an ultimately-periodic continued fraction. So the only thing that is interesting is the initial segment of the CF before the period begins. That initial segment contains a long string of 1s basically because the Plouffe numbers when k is large yield very good approximations of the golden ratio. But the funny thing is, it then looks like a periodic sequence begins, but then it isn't what you thought, then it looks like a longer periodic sequence begins, but it again isn't what you thought, and so on for quite a while. So it is as though Plouffe's quadratic irrational, is very well-approximated by a sequence of less-complex quadratic irrationals. Well, that actually is (of course) true, although it may not be obvious which precise subsequence among them are good enough to cause this behavior. On 11/12/15, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:

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Today's Topics:

1. about some continued fractions related to sqrt(5). (Simon Plouffe) 2. Re: about some continued fractions related to sqrt(5). (Mike Stay) 3. Re: about some continued fractions related to sqrt(5). (Dan Asimov) 4. Re: about some continued fractions related to sqrt(5). (Dan Asimov) 5. Re: about some continued fractions related to sqrt(5). (Dan Asimov)

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Message: 1 Date: Thu, 12 Nov 2015 20:30:14 +0100 From: Simon Plouffe <simon.plouffe@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] about some continued fractions related to sqrt(5). Message-ID: <5644E8C6.1020109@gmail.com> Content-Type: text/plain; charset=utf-8; format=flowed

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

------------------------------

Message: 2 Date: Thu, 12 Nov 2015 12:07:15 -0800 From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] about some continued fractions related to sqrt(5). Message-ID: <CAKQgqTavya1y3Uci+r8iNL1FV-cTQn5Va6xjHJku33fGYeFDpQ@mail.gmail.com> Content-Type: text/plain; charset=UTF-8

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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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

------------------------------

Message: 3 Date: Thu, 12 Nov 2015 12:27:01 -0800 From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] about some continued fractions related to sqrt(5). Message-ID: <4A46CFC5-64BD-4738-8B8D-A89E8E20C4B9@earthlink.net> Content-Type: text/plain; charset=utf-8

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

------------------------------

Message: 4 Date: Thu, 12 Nov 2015 12:32:03 -0800 From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] about some continued fractions related to sqrt(5). Message-ID: <B66BFE30-3B45-4A90-A173-3F5A43A25868@msri.org> Content-Type: text/plain; charset=utf-8

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

------------------------------

Message: 5 Date: Thu, 12 Nov 2015 14:35:25 -0800 From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] about some continued fractions related to sqrt(5). Message-ID: <169CB4ED-D4EC-4131-8A69-B6629F3CC329@earthlink.net> Content-Type: text/plain; charset=utf-8

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.

------------------------------

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