Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$

Question

I know how to prove the fact that $ p$ | ${ p \choose k } $ (when writing it as a fraction, $p$ cannot be divided by any of the $1\times2\times...\times k$ or $1\times2\times...\times(p-k)$ because $p$ is prime).

When I try to apply the same rationale for $ p$ | ${ p^n \choose k } $ I get stumped because $p^n$ is in fact divisible by one of the $k! = 1\times2\times...\times k$, where $0 < k < p^n$.

If we take the example of $2^3$ and $k = 7$, we can easily see that $k! = 1 \times 2 \times ... \times 6 \times 7$, and $2$ clearly divides $2^3$.

How else can I approach this?

Answer 1

The following identity holds:

$$\binom{p^n}{k}=\frac{p^n}{k} \binom{p^n-1}{k-1}.$$

Hence

$$k\binom{p^n}{k}=p^n\binom{p^n-1}{k-1}$$

which implies that $$p^n\mid k\binom{p^n}{k}.$$

• If $p$ and $k$ are coprime, then $p^n \mid \binom{p^n}{k}$, with $p \mid p^n$. This completes the proof.

• If they're not, then there exists a maximal $\alpha <n$ and $q \in \mathbb N$ such that $k=p^\alpha q$ (note that $q$ and $p$ are coprime).

We have that $$q \binom{p^n}{k}=p^{n-\alpha} \binom{p^n-1}{k-1}$$ and $n-\alpha \geq 1$

Thus $$p\mid p^{n-\alpha}$$ and $$p^{n-\alpha}\mid \binom{p^n}{k}.$$

Finally $$p\mid \binom{p^n}{k}.$$

Answer 2

The highest prime power $p$ dividing $n!$ is expressed as $$\sum_{x=1}^\infty \lfloor \frac{n}{p^x} \rfloor.$$ Since $\binom{p^n}{k}=\frac{(p^n)!}{k!(p^n-k)!},$ we compare the highest prime power $p$ which divide numerator and denominator. $$v_p((p^n)!)=\sum_{x=1}^n \lfloor \frac{p^n}{p^x} \rfloor,$$ $$v_p(k!(p^n-k)!)=v_p(k!)+v_p((p^n-k)!)=\sum_{x=0}^n \left( \lfloor \frac{k}{p^x} \rfloor + \lfloor \frac{p^n-k}{p^x} \rfloor \right).$$ (We've replaced $\infty$ bound with $n$ since if $x>n,$ the terms evaluate to $0$).

Notice that $\lfloor a+b \rfloor \ge \lfloor a \rfloor + \lfloor b \rfloor,$ and the that if $a+b$ is an integer while $a$ and $b$ are not, then $\lfloor a+b \rfloor = \lfloor a \rfloor + \lfloor b \rfloor - 1.$ We can conclude with the first fact that $$\sum_{x=1}^n \lfloor \frac{p^n}{p^x} \rfloor \ge \sum_{x=0}^n \left( \lfloor \frac{k}{p^x} \rfloor + \lfloor \frac{p^n-k}{p^x} \rfloor \right).$$

Because

1. $0<k<p^n,$

2. $\frac{p^n}{p^x} \in \mathbb{Z}$ for integers $1 \le x \le n,$

3. $\frac{k}{p^x}+\frac{p^n-k}{p^x} \in \mathbb{Z}$ for same bounds,

4. When $x=n,$ neither $\frac{k}{p^x}$ nor $\frac{p^n-k}{p^x}$ are integral $\implies \lfloor \frac{p^n}{p^n} \rfloor = 1 > \lfloor \frac{k}{p^n} \rfloor + \lfloor \frac{p^n-k}{p^n} \rfloor = 0.$

Finally, we can conclude $$\sum_{x=1}^n \lfloor \frac{p^n}{p^x} \rfloor > \sum_{x=0}^n \left( \lfloor \frac{k}{p^x} \rfloor + \lfloor \frac{p^n-k}{p^x} \rfloor \right).$$ This implies the divisibility.

Answer 3

Here is a proof which uses ring theory. Let $F=\mathbb Z/p\mathbb Z$ where $p$ is a prime.

Consider the polynomial ring $R=F[X,Y]$. The map $\varphi:R\to R$ given by $\varphi(a)=a^p$ is a ring homomorphism. This follows from the observation that $p$ divides $\binom{p}{k}$ for every $1\leq k\leq p-1$.

Let $m$ be a positive integer. As $\varphi$ is a ring homomorphism, it follows that $\varphi^m$ is also a ring homomorphism. Thus, we have that $\varphi(X+Y)=\varphi(X)+\varphi(Y)$ and hence $(X+Y)^{p^m}=X^{p^m}+Y^{p^m}$. So it follows that $p$ divides $\binom{p^m}{k}$ for every $1\leq k\leq p^m-1$.

Answer 4

Edit: The question was changed after the answer below was posted.

In general, it is not true that $p^n$ divides $\binom{p^n}{k}$. Let $p=2$, $n=2$, $k=2$. There are infinitely many examples.

Answer 5

This answer was started before the question was changed.

Note that the highest power of $p$ which divides $n!$ is $$\sum_{r=1}^\infty\left\lfloor\frac {n}{p^r}\right\rfloor$$ which is a finite sum because the terms eventually become zero.

The highest power of $p$ which divides $(p^n)!$ is $p^{n-1}+p^{n-2}+\dots 1=\cfrac {p^n-1}{p-1}$

The highest power of $p$ which divides $(kp^{n-1})!$, where $1\le k \lt p$, is $kp^{n-2}+kp^{n-3}+\dots +k=k\cfrac{p^{n-1}-1}{p-1}$

Now examine $$\binom {p^n}{kp^{n-1}}$$ Subtracting the two contributions made by the numerator from that made by the denominator, the total power of $p$ dividing this is $$\cfrac {p^n-1}{p-1}-k\cfrac {p^{n-1}-1}{p-1}-(p-k)\cfrac {p^{n-1}-1}{p-1}=\frac {p^n-1-kp^{n-1}-(p-k)p^{n-1}+k+(p-k)}{p-1}=1$$

This shows that the highest possible power of $p$ is always $1$.

This technique can be used to show that a power of $p$ will always divide the coefficient (except the ones which are $1$) and to analyse what power of $p$ that will be.