Binomial coefficients $\binom{n}{b} \pmod{n}$, $n > b$, $n$ odd, $n,b$ positive integers, AKS test

Question

I am looking at the values of Binomial coefficients $\binom{n}{b} \pmod{n}$, where $n > b$, $n$ odd, and $n,b$ positive integers. I am planning to use the results in an implementation in python of the AKS primality test.

Examples of binimial coeffs mod $n$:

1. $b=3, n=5$, $$\binom{5}{3} \pmod{5} \cong 0$$

2. $b=10, n=15$, $$\binom{15}{10} \pmod{15} \cong 3$$

Since it is mentioned in comments, Lucas's theorem is for $n$ prime, but I am considering odd $n$.

I have found that there are several patterns, that I think may depend on the factorisation of $b$

Examples of patterns found (using assumptions stated):

1. For $b=6$:

$\binom{n}{6} \pmod{n} \cong \frac{n}{3}$ when $n \pmod{3} \cong 0$ and $n \pmod{9} \cong 0$

$\binom{n}{6} \pmod{n} \cong \frac{2n}{3}$ when $n \pmod{3} \cong 0$ and $n \pmod{9} \cong 6$

$\binom{n}{6} \pmod{n} \cong 0$ when $n \pmod{3}$ is not cong to $0$

2. For $b=7$:

$\binom{n}{7} \pmod{n} = \frac{n}{7}$ when $n \pmod{7} \cong 0$

$\binom{n}{7} \pmod{n} = 0$ when $n \pmod{7} $ not cong to $0$

3. For $b=8$:

$\binom{n}{8} \pmod{n} \cong 0$

Questions:

1. What is the relationship between $\binom{n}{b} \pmod{n}$ and $b$ for odd $n$?

2. Seeking reference requests.

Answer 1

As a consequence of theorem 4 in

1. Andrew D. Loveless: A Congruence for Products of Binomial Coefficients modulo a Composite, in: INTEGERS 7 (A44 10 Jan 2007)

we have $\binom{n}{b}\equiv 0\pmod{n}$ if (but not only if) $\gcd(n,b)=1$. The other theorems and corollaries in the paper may be of interest to you as well.