Mathematics – Number Theory
Scientific paper
2007-09-05
"Gems in Experimental Mathematics" (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Am
Mathematics
Number Theory
16 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
Scientific paper
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.
Schalm Kyle
Sondow Jonathan
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