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This is an expanded version of our earlier paper. Let the $n$th partial sumof 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 ourconjecture that only two of the partial sums are convergents to $e$. A relatedresult about the denominators $q_k$ and powers of factorials is proved. We alsoshow 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 asecond conjecture we make about the zeros of $A_n$ and $q_k$ modulo powers of2. Tables supporting this Zeros Conjecture are presented and we discuss a2-adic reformulation of it.

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