Some Congruences for Binomial Coefficients

Abstract

Suppose that $p = tn + r$ is a prime and that $h$ is the class number of the imaginary quadratic field, $\mathbb{Q}(\sqrt{-t})$. If $t \equiv 3$ (mod 4) is a prime, just $r$ is a quadratic residue modulo $t$ and the order of $r$ modulo $t$ is $\frac{t-1}{2}$, then $4p^h$ can be written in the form $a^2 + tb^2$ for some integers $a$ and $b$. And if $t = 4k$ where $k \equiv 1$ (mod 4), $r \equiv 3$ (mod 4), $r$ is a quadratic non-residue modulo $t$ and the order of $r$ modulo $t$ is $k - 1$, then $p^h = a^2 + kb^2$ for some integers $a$ and $b$. Our result is that $a$ or $2a$ is congruent modulo $p$ to a product of certain binomial coefficients modulo sign. As an example, we give explicit formulas for $t = 11$, $19$, $20$ and $23$.