Absolute irreducibility of the binomial polynomials

TL;DR

This paper proves that binomial polynomials are absolutely irreducible in the ring of integer-valued polynomials, meaning their powers factor uniquely into irreducibles, using linear algebra and number theory techniques.

Contribution

It establishes the absolute irreducibility of binomial polynomials in $ ext{Int}( ext{Z})$, filling a gap in understanding their factorization behavior.

Findings

Binomial polynomials are absolutely irreducible in $ ext{Int}( ext{Z})$.

Factorization of their powers is unique into irreducibles.

A key number-theoretic result about primes dividing consecutive composite integers.

Abstract

In this paper we investigate the factorization behaviour of the binomial polynomials $\binom{x}{n} = \frac{x(x-1)\cdots (x-n+1)}{n!}$ and their powers in the ring of integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. While it is well-known that the binomial polynomials are irreducible elements in $\operatorname{Int}(\mathbb{Z})$, the factorization behaviour of their powers has not yet been fully understood. We fill this gap and show that the binomial polynomials are absolutely irreducible in $\operatorname{Int}(\mathbb{Z})$, that is, $\binom{x}{n}^m$ factors uniquely into irreducible elements in $\operatorname{Int}(\mathbb{Z})$ for all $m\in \mathbb{N}$. By reformulating the problem in terms of linear algebra and number theory, we show that the question can be reduced to determining the rank of, what we call, the valuation matrix of $n$. A main ingredient in computing this rank is the following number-theoretical result for which we also provide a proof: If $n>10$ and $n$, $n-1$, \ldots, $n-(k-1)$ are composite integers, then there exists a prime number $p > 2k$ that divides one of these integers.