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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:
11J72Irrationality
11A25Arithmetic functions, etc.