Diophantine approximation of the exponential function and Sondowʼs Conjecture

Abstract

We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of e2/a published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the convergents of e and prove a conjecture of J. Sondow that there are only two instances when the convergents of the continued fraction of e coalesce with partial sums of e. We conclude with open questions about the zeros of certain p-adic functions naturally occurring in our proofs.