[math-fun] Continued fraction for exp(3)
What is known about the simple continued fraction expansion of exp(3)? Obviously we can generate it fairly easily, but I want to know if there are any known patterns or periodicities in the terms. I thought exp(3) was a known Hurwitz number (continued fraction consists of interleaved polynomials) but I can't see any obvious patterns.
"However there is no known formula for the partial quotients of the continued fraction expansion of e^3, or more generally e^{l/m}, with l distinct from 1,2 and gcd(l,m)=1." http://www.numbertheory.org/php/davison.html On Wed, Sep 5, 2012 at 12:17 PM, Allan Wechsler <acwacw@gmail.com> wrote:
What is known about the simple continued fraction expansion of exp(3)? Obviously we can generate it fairly easily, but I want to know if there are any known patterns or periodicities in the terms. I thought exp(3) was a known Hurwitz number (continued fraction consists of interleaved polynomials) but I can't see any obvious patterns. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://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
As far as I know, nothing is known. I computed thousands of partial quotients 30 years ago, but there was no simple pattern I could detect. On 9/5/12 3:17 PM, Allan Wechsler wrote:
What is known about the simple continued fraction expansion of exp(3)? Obviously we can generate it fairly easily, but I want to know if there are any known patterns or periodicities in the terms. I thought exp(3) was a known Hurwitz number (continued fraction consists of interleaved polynomials) but I can't see any obvious patterns. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
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Allan Wechsler -
Jeffrey Shallit -
Mike Stay