Abstract factorials

Angelo B. Mingarelli
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 4, Pages 43–76
Full paper (PDF, 301 Kb)

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Authors and affiliations

Angelo B. Mingarelli
School of Mathematics and Statistics
Carleton University, Ottawa, Ontario, Canada, K1S 5B6

Abstract

A commutative semigroup of abstract factorials is defined in the context of the ring of integers. Given a subset X ⊆ ℤ+, or ℤ, we construct a “factorial set” with which one may define a multitude of abstract factorials on X. These factorial sets are then used to show that given any set X of positive integers we can define infinitely many factorial functions on X each having interesting properties. Such factorials are also studied independently of whether or not there is an association to sets of integers. Using an axiomatic approach we study the possible equality of consecutive factorials, the ratios of consecutive factorials and we provide many examples outlining the applications of the ensuing theory; examples dealing with sets of prime numbers, Fibonacci numbers, and highly composite numbers among other sets of integers. One of the advantages in using this setting is that many apparently independent irrationality criteria involving factorials can be assimilated within this scheme.

Keywords

• Abstract factorial
• Factorial
• Factorial sets
• Factorial sequence
• Irrational
• Divisor function
• Von Mangoldt function
• Cumulative product
• Hardy–Littlewood conjecture
• Prime numbers
• Highly composite numbers
• Fibonacci numbers

AMS Classification

• Primary: 11B65, 11A25, 11J72
• Secondary: 11A41, 11B39, 11B75, 11Y55

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