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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
##### Keywords:
irrationality measures; arithmetic functions
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