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

*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

${s}_{n}\xb7n!$, whose truth would imply the truth of the first conjecture.

##### MSC:

11A55 | Continued fractions (number-theoretic results) |

11B50 | Sequences (mod $m$) |

11B83 | Special sequences of integers and polynomials |

11J70 | Continued fractions and generalizations |

11J82 | Measures of irrationality and of transcendence |

11Y55 | Calculation of integer sequences |