Which Partial Sums of the Taylor Series for $e$ are Convergents to $e$? (and a Link to the Primes 2, 5, 13, 37, 463), II

Abstract

This is an expanded version of our earlier paper. Let the $n$th partial sum of the Taylor series $e={\sum }_{r=0}^{\infty }1/r!$ be $A_n/n!$, and let $p_k/q_k$ be the $k$th convergent of the simple continued fraction for $e$. Using a recent measure of irrationality for $e$, we prove weak versions of our conjecture that only two of the partial sums are convergents to $e$. A related result about the denominators $q_k$ and powers of factorials is proved. We also show a surprising connection between the $A_n$ and the primes 2, 5, 13, 37, 463. In the Appendix, we give a conditional proof of the conjecture, assuming a second conjecture we make about the zeros of $A_n$ and $q_k$ modulo powers of 2. Tables supporting this Zeros Conjecture are presented and we discuss a 2-adic reformulation of it.

arXiv comment16 pages, 6 tables. In the new Section 4, we prove that if p/q is a convergent to e with q = (n!)^k for some n and k > 0, then n/k < e. The old Section 4 is now Section 5. Several parts of the Appendix are clarified. To appear in ``Tapas in Experimental Mathematics 2009'' in the AMS series Contemporary Mathematics