Binomial Coefficient 2 n Choose n is Divisible by All Primes between n and 2 n

Theorem

Let $\dbinom {2 n} n$ denote a binomial coefficient.

Then for all prime numbers $p$ such that $n < p < 2 n$:

$p \divides \dbinom {2 n} n$

where $\divides$ denotes divisibility.

Proof

By definition of binomial coefficient:

\(\ds \dbinom {2 n} n\)

\(=\)

\(\ds \dfrac {\paren {2 n}!} {\paren {n!}^2}\)

where $n!$ denotes the factorial of $n$

\(\ds \leadsto \ \ \)

\(\ds \dbinom {2 n} n \paren {n!}^2\)

\(=\)

\(\ds \paren {2 n}!\)

Let $\mathbb P$ denote the set of prime numbers.

Let $p \in \mathbb P$ such that $n < p < 2 n$.

By definition of factorial:

$p \divides \paren {2 n}!$

That is:

$p \divides \dbinom {2 n} n \paren {n!}^2$

From Prime iff Coprime to all Smaller Positive Integers:

$p \nmid n!$

where $\nmid$ denotes non-divisibility.

and so:

$p \nmid \paren {n!}^2$

But from Euclid's Lemma for Prime Divisors, $p$ is a divisor of either $\dbinom {2 n} n$ or $\paren {n!}^2$.

Hence it must be the case that:

$p \divides \dbinom {2 n} n$

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.2$: Divisibility and factorization in $\mathbf Z$: Exercise $3$