Absolute irreducibility of the binomial polynomials

Roswitha Rissner, Daniel Windisch

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.