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A geometric proof that \(e\) is irrrational and a new measure of its irrationality. (English) Zbl 1149.11035
Let \(S(n)\) denote the Kempner-Smarandache arithmetic function defined by \(S(n)= \min\{k> 0\): \(n\) divides \(k!\}\). The author proves the following interesting measure of irrationality for the number \(e\), involving \(S(n)\): for all integers \(p\) and \(q\), with \(q> 1\), one has \(|e- p/q|> 1/(S(q)+ 1)\)! From the two conjectures stated in this paper, we mention the following one: the inequality \(q^2< S(q)!\) holds for almost all \(q\).

MSC:
11J72 Irrationality; linear independence over a field
11A25 Arithmetic functions; related numbers; inversion formulas
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