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Which partial sums of the Taylor series for $e$ are convergents to $e$? (and a link to the primes 2, 5, 13, 37, 463). (English) Zbl 1159.11004
Amdeberhan, Tewodros (ed.) et al., Tapas in experimental mathematics. AMS special session on experimental mathematics, New Orleans, LA, USA, January 5, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4317-8/pbk). Contemporary Mathematics 457, 273-284 (2008).
Let $s_n:=\sum_{r=0}^n \frac{1}{r!}$ be the $n$-th partial sum of the Taylor series for $e$. The authors conjecture that the question posed in the title of this paper has the answer “Only $s_1$ and $s_3$”. They prove a weak form of this conjecture, namely that asymptotically almost surely the partial sums are {\it not} convergents $p/q$ to the simple continued fraction expansion of $e$. They also present (and give experimental evidence for) a second conjecture about the periodic behaviour modulo powers of 2 of the denominators of the convergents, $q$, and of the quantities $s_n \cdot n!$, whose truth would imply the truth of the first conjecture. For the entire collection see [Zbl 1139.00009].

11A55Continued fractions (number-theoretic results)
11B50Sequences (mod $m$)
11B83Special sequences of integers and polynomials
11J70Continued fractions and generalizations
11J82Measures of irrationality and of transcendence
11Y55Calculation of integer sequences