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Diophantine approximation of the exponential function and Sondow’s conjecture. (English) Zbl 1292.11079

This paper starts with examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of \(e^{2/a}\) 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 \(e\). The authors then develop a \(p\)-adic analysis of the denominators of the convergents of \(e\) and prove a conjecture of J. Sondow [Am. Math. Mon. 113, No. 7, 637–641 (2006; Zbl 1149.11035)] that there are only two instances when the convergents of the continued fraction of \(e\) coalesce with partial sums of \(e\). The authors conclude with open questions about the zeros of certain \(p\)-adic functions naturally occurring in their proofs.

MSC:

11J70 Continued fractions and generalizations
11A55 Continued fractions
11B50 Sequences (mod \(m\))

Citations:

Zbl 1149.11035

Software:

OEIS
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Full Text: DOI

References:

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