Re: [math-fun] Pi to 31.4 trilllion digits
What might be interesting would be to see kajillions of "digits" of the continued fraction expansion (CFE) of π. At least, that's independent of how many fingers we have. Are there good algorithms for calculating a high "digit" of the CFE of π without knowing the previous ones? —Dan Simon Plouffe schrieb: ----- Pi has been calculated to 31.4 trillion digits : http://www.numberworld.org/blogs/2019_3_14_pi_record/ they used the Chudnovsky formula, the Bellard formula and mine just to be certain. -----
I am guessing that an algorithm to extract the nth term for the continued fraction of pi is *harder* than an algorithm to extract the nth digit. Consider: The first term of the continued fraction of x is floor(x). The second term is floor(1/(x-floor(x))). Can one calculate that directly, without calculating floor(x) first? It doesn't seem likely. On Fri, Mar 15, 2019 at 3:25 PM Dan Asimov <dasimov@earthlink.net> wrote:
What might be interesting would be to see kajillions of "digits" of the continued fraction expansion (CFE) of π. At least, that's independent of how many fingers we have.
Are there good algorithms for calculating a high "digit" of the CFE of π without knowing the previous ones?
—Dan
Simon Plouffe schrieb: ----- Pi has been calculated to 31.4 trillion digits : http://www.numberworld.org/blogs/2019_3_14_pi_record/
they used the Chudnovsky formula, the Bellard formula and mine just to be certain. -----
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The equivalent question for e is easy enough. --ms On 15-Mar-19 16:02, Allan Wechsler wrote:
I am guessing that an algorithm to extract the nth term for the continued fraction of pi is *harder* than an algorithm to extract the nth digit. Consider: The first term of the continued fraction of x is floor(x). The second term is floor(1/(x-floor(x))). Can one calculate that directly, without calculating floor(x) first? It doesn't seem likely.
On Fri, Mar 15, 2019 at 3:25 PM Dan Asimov <dasimov@earthlink.net> wrote:
What might be interesting would be to see kajillions of "digits" of the continued fraction expansion (CFE) of π. At least, that's independent of how many fingers we have.
Are there good algorithms for calculating a high "digit" of the CFE of π without knowing the previous ones?
—Dan
Simon Plouffe schrieb: ----- Pi has been calculated to 31.4 trillion digits : http://www.numberworld.org/blogs/2019_3_14_pi_record/
they used the Chudnovsky formula, the Bellard formula and mine just to be certain. -----
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It depends. There are generalized continued fractions where it's very easy, like π = 3 + 1^2 / (6 + 3^2 / (6 + 5^2 / (6 + 7^2 ( 6 + ... ) ) ) ) Nobody knows a spigot-like algorithm for the simple continued fraction of pi. On Fri, Mar 15, 2019 at 1:25 PM Dan Asimov <dasimov@earthlink.net> wrote:
What might be interesting would be to see kajillions of "digits" of the continued fraction expansion (CFE) of π. At least, that's independent of how many fingers we have.
Are there good algorithms for calculating a high "digit" of the CFE of π without knowing the previous ones?
—Dan
Simon Plouffe schrieb: ----- Pi has been calculated to 31.4 trillion digits : http://www.numberworld.org/blogs/2019_3_14_pi_record/
they used the Chudnovsky formula, the Bellard formula and mine just to be certain. -----
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-- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com
Hello, as you may know, only the quadratic irrationals are predictable, (Lagrange theorem), the others being the exponential and some variants as well as some points with the Bessel function. I am afraid there is nothing else new in that area since approx. 1850. But... there is a pattern if we look at the generalized continued fraction of Pi, (or 1/Pi) . As far as I know : we know nothing from the CFE of the cube root of 2 either. There is a pattern in the CFE of Pi but it is with the generalized expansion of the continued fraction. It is more general but we lose the convergence property. And that's all that is known for this subject, nothing better has been found since Euler and the 1850s thereafter. Euler's traditional transformation between continuous fractions has not advanced an inch since. I have a program that expands a real number into 1000 different developments, I put all these transformations in my big table of 17.3 billion constants and it has not given anything until now. It is mainly based on variants of the greedy algorithm. Unless someone finds better than what Euler found: we're stuck. In other words , I have no idea where to look beyond : Pi = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, ...] ?? Best regards, Simon Plouffe
participants (5)
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Allan Wechsler -
Dan Asimov -
Mike Speciner -
Mike Stay -
Simon Plouffe