#### Finding the continued fraction of *e*^{p/q}

The continued fraction expansions for *e*^{1/k} and *e*^{2/k} (k odd) are well known, namely:

However there is no known formula for the partial quotients of the continued fraction expansion of *e*^{3}, or more generally *e*^{p/q}, with *p* distinct from 1,2 and gcd(*p,q*)=1.

Here are the first 250,000 partial quotients of e^{3} and here is the table size-sorted. The largest partial quotient is a_{196440} = 546341.
We perform the algorithm of J.L. Davison, *An algorithm for the continued fraction of e*^{p/m}, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, 1978), 169-179, Congress. Numer., XXII, Utilitas Math.

The starting point is a result of R.F.C. Walters in *Alternate derivation of some regular continued fractions*, J. Austr. Math. Soc 8 (1968), 205-212): If

then *p*_{n}/r_{n} and *q*_{n}/s_{n} → *e*^{l/m}.

We first find the least n=n* such that *p*_{n}, q_{n}, r_{n}, s_{n} are non-negative and repeatedly apply Raney's factorisation for n*≤ k ≤ n*+N, as in Davison's example in §3.

The number (*count*) of partial quotients of *e*^{p/q} found is returned.

We cannot predict the value of *count*, but it becomes positive for sufficiently large N.

We restrict taking p/q no greater than 14, because of the time taken to compute a_{0} for larger values of p/q.

This is a BCMATH translation of a BC program.

A faster C version is available.

Inside the program, *p* and *q* are replaced by *p/gcd(p,q)* and *q/gcd(p,q)*.

*Last modified 19th August 2017*

Return to main page