Abstract factorial functions and their applications

Details Abstract factorial functions and their applications Abstract factorial functions and their applications

2007-05-29

arxiv.org/abs/0705.4299v2

Mathematics

Number Theory

21 pages, preprint; comments welcome

Scientific paper

We define the notion of an abstract factorial function on the set of natural numbers and show that, given any subset of the integers, we can associate to it another set with which, if non-trivial, we can define one or more (generally independent) abstract factorial functions. These associated sets are studied and arithmetic relations are revealed. In particular, we show that one of the associated sets of the set of primes is a subset of a class of numbers that also contains the highly composite numbers of Ramanujan. Furthermore, we show that for any given abstract factorial function the series of reciprocals of its factorials always converges to an irrational number.

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