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

TL;DR

This paper investigates which partial sums of the Taylor series for e are convergents to e, providing partial proofs, conjectures, and connections to prime numbers and 2-adic properties, advancing understanding of e's approximation.

Contribution

It proves weak versions of the conjecture that only two partial sums are convergents to e, and establishes a surprising link between these sums and specific primes.

Findings

Weak proof that only two partial sums are convergents to e

Connection between partial sums and primes 2, 5, 13, 37, 463

Conditional proof of the conjecture based on zeros of sequences modulo powers of 2

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.