[Paper Review] Abstract factorials

Angelo B. Mingarelli|arXiv (Cornell University)|May 29, 2007

Rings, Modules, and Algebras6 references4 citations

TL;DR

This paper introduces a general framework for abstract factorials on subsets of positive integers, defining them via factorial sets and proving that the sum of reciprocals of any such abstract factorial always converges to an irrational number. The theory applies broadly, establishing irrationality for series involving primes, Fibonacci numbers, highly composite numbers, and divisor functions, using a semigroup structure and concavity conditions on generalized factorial functions.

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.

Motivation & Objective

• To define a general class of abstract factorials on subsets of positive integers using factorial sets, extending beyond classical factorials.
• To establish that the sum of reciprocals of any abstract factorial sequence is always irrational, regardless of the underlying set.
• To develop a semigroup structure on abstract factorials to derive irrationality results for combined factorial sequences.
• To provide a unifying framework that includes known generalized factorials (e.g., Bhargava's) and extends to new classes like divisor functions and Fibonacci products.
• To prove irrationality for specific series involving prime numbers, highly composite numbers, and cumulative divisor products.

Proposed method

• Define an abstract factorial as a map $!_a: \mathbb{N} \to \mathbb{Z}^+$ satisfying minimal conditions, ensuring closure under multiplication and compatibility with binomial coefficients.
• Construct factorial sets from any subset $X \subseteq \mathbb{Z}^+$, enabling the generation of infinitely many abstract factorials from a single set.
• Use the semigroup property of abstract factorials to prove irrationality of sums of the form $\sum_{n=0}^\infty \frac{1}{\prod_{j=1}^k n!_j^{s_j}}$ with $s_j \in \mathbb{N}$, not all zero.
• Apply a dichotomy on the limit superior of ratios of consecutive factorials to analyze growth behavior and avoid infinite runs of equal factorials.
• Leverage concavity conditions on functions $f: \mathbb{Z}^+ \times \mathbb{Z}^+ \to \mathbb{Z}^+$, such as $f(n,q) = \binom{n+q-1}{q}$, to ensure integrality of generalized binomial coefficients and prove irrationality of related series.
• Use contradiction arguments based on integral approximations of partial sums to prove irrationality, particularly showing $\alpha_k < 1$ when assuming rationality of the full sum.

Experimental results

Research questions

• RQ1 Can a general theory of abstract factorials be constructed on arbitrary subsets of positive integers that generalizes classical and known generalized factorials?
• RQ2 Under what conditions does the sum of reciprocals of an abstract factorial sequence converge to an irrational number?
• RQ3 Can the semigroup structure of abstract factorials be used to prove irrationality of combined factorial series?
• RQ4 What role does the concavity of the function defining the factorial growth play in ensuring integrality of generalized binomial coefficients?
• RQ5 Do specific sequences like primes, Fibonacci numbers, or highly composite numbers give rise to abstract factorials whose reciprocal series are irrational?

Key findings

• The sum of reciprocals of any abstract factorial sequence always converges to an irrational number, regardless of the underlying set or construction.
• For any positive integers $b, q, k$, the series $\sum_{n=1}^\infty \frac{1}{n! \, q^{\sum_{j=1}^n d(j)}}$ is irrational, where $d(j)$ is the number of divisors of $j$.
• The series $\sum_{n=1}^\infty \frac{1}{p_n!}$, where $p_n$ is the $n$-th prime, is irrational, as is $\sum_{n=1}^\infty \frac{1}{\mathcal{F}(n)^k}$ with $\mathcal{F}(n)$ the product of the first $n$ Fibonacci numbers.
• The series $\sum_{n=1}^\infty \frac{1}{q^{f(n,q)} \, n!}$ is irrational for any $q \in \mathbb{Z}^+$ and any function $f$ satisfying $f(x+y,q) \geq f(x,q) + f(y,q)$, including $f(n,q) = \binom{n+q-1}{q}$.
• The series $\sum_{n=1}^\infty \frac{1}{\prod_{i=1}^n h_i^{\lfloor n/i \rfloor}}$ is irrational when $h_i$ are highly composite numbers, due to the self-factorial property of their factorial set.
• The generalized binomial coefficients $\binom{n}{k}_a = \left(\frac{n!}{k!}\right)^{qk} \cdot \left(\frac{n!}{(n-k)!}\right)^{q(n-k)}$ are integers when $n!_a = n!^{qn}$, ensuring the validity of the abstract factorial framework.