Abstract factorials

TL;DR

This paper explores the theory of abstract factorials within the ring of integers, constructing factorial sets for subsets of positive integers, analyzing their properties, and demonstrating that reciprocal series always converge to irrational numbers.

Contribution

It introduces the concept of factorial sets for integer subsets, studies properties of abstract factorials, and proves the irrationality of reciprocal series for these factorials.

Findings

Reciprocal series of abstract factorials converge to irrational numbers.

Constructs factorial sets for various integer subsets like primes and Fibonacci numbers.

Analyzes properties such as equality of consecutive factorials and ratio limits.

Abstract

A commutative semigroup of abstract factorials is defined in the context of the ring of integers. We study such factorials for their own sake, whether they are or are not connected to sets of integers. Given a subset X of the positive integers we construct a "factorial set" with which one may define a multitude of abstract factorials on X. We study the possible equality of consecutive factorials, a dichotomy involving the limit superior of the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of our results states that given any abstract factorial the series of reciprocals of its factorials always converges to an irrational number. Thus, for example, for any positive integer k the series of the reciprocals of the k-th powers of the cumulative product of the divisors of the numbers from 1 to n is irrational.