I was programming the AKS primality test, and I have two questions.

Is it correct that if gcd(n, k)=1, then binomial(n, k)=0 (mod n)?

Is it correct that if gcd(n, k)>1, then binomial(n, k)>0 (mod n)?

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    $\begingroup$ This looks a math question rather than a CS one. (use then if not comparing) $\endgroup$ – greybeard Jul 15 '19 at 4:12

All variables are understood to be positive integers. Let $n\mid m$ mean $n$ is a factor of $m$, a.k.a. $m$ is divisible by $n$ or $m=0\pmod n$.

Is it correct that if $\gcd(n, k)=1$, then $n\mid\binom nk$?


Since $k\binom nk=n\binom{n-1}{k-1}$, we know $k\binom nk$ is divisible by $n$. Since $\gcd(n,k)=1$, we must have $\binom nk$ is divisible by $n$.

Is it correct that if $\gcd(n, k)>1$, then $\binom nk>0\pmod n$?

The relation of "greater than" or "less than" modulo $n$ does not make sense since, for example, $-1=1=2019\pmod 2$.

What you have in mind should be the following question.

Is it correct that if $\gcd(n, k)>1$, then $n\not\mid\binom nk$?

Not necessarily.

The smallest counterexample is $\gcd(10,4)=2\neq1$ but $10\mid\binom{10}4=210$. In fact, if $n=2\pmod8$, then $n\mid\binom n4$.

Kummer's theorem provides a way to compute the exponent of the highest power of a given prime number dividing a binomial coefficient. Peter Taylor points out that it can be used to determine whether $n\mid\binom nk$ in polynomial time if we know the prime factorisation of $k$ or $n$. However, no algorithm is known to be able to obtain the prime factorisation of an integer in polynomial-time.

Here is a related interesting article, divisibility properties of binomial coefficients by K. R. Mclean.

Exercise 1. Show that if $m\mid n$ and $\gcd(m,k)=1$, then $m\mid\binom nk$.

Exercise 2. Show that $2i\mid\binom{2i}4$ iff $i=1\pmod4$.

Exercise 3. Find infinitely many pairs of $n$ and $k$ such that $1\lt k \mid n$ and $n\mid\binom nk$. (Hint, one such family can be found by setting $k=6$.)

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