#### Finding the regular continued fraction of a rational multiple *m/n* of *e*^{1/q}

The algorithm is described in the paper *Some properties of the continued fraction expansion of (m/n)e*^{1/q}, K.R. Matthews and R.F.C. Walters, Proc. Camb. Phil. Soc. (1970), **67**, 67-74. It uses a version of Satz 4.1. of *Die Lehre von den Kettenbrüchen*, Band 1, S. 111.
Inside the program, *m* and *n* are replaced by *m'=m/gcd(m,n)* and *n'=n/gcd(m,n)*.
We limit input to *m'n'* ≤ *2000*.

The paper shows that there is a quasi-period consisting of d=mn linear expressions of the form p_{t}^{2}2qx+q_{t}, t=1,...,d. A criteria is given for all the p_{t} to be equal to 1. In the case of (m/1)e, the primes m = 7, 11, 13, 31, 41, ... give at least one progression with p_{t} > 1.

In the first version of this program, a zero partial quotient was present, being located at the end of the factorizations of the first or the start of the factorization of the second matrix in Theorem 1 (b) of the paper. This is now
removed, thanks to an elegant proof of
Alan Offer on 30th June 2010. We also have improved the code for getting the "initial" matrix, by using a suggestion of Alan that we use the recurrence relations of his proof.

*Last modified 9th January 2013*

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