Two \(p^ 3\) variations of Lucas' theorem
The purpose of this paper is to give two variations of Lucas' theorem which established a method of computing binomial coefficients modulo a prime. The following lemma is the key to the main results: If p is a prime and \(p\geq 5\), then \(\left( \begin{matrix} kp\\ p\end{matrix} \right)=k (mod p^ 3).\) The main result is as follows. If k and r are non-negative integers, p is a prime, and \(p\geq 5\), then \[ \left( \begin{matrix} kp\\ rp\end{matrix} \right)=\left( \begin{matrix} k\\ r\end{matrix} \right)(mod p^ 3). \] The method used to prove the above two results is by induction.
scientific article; zbMATH DE number 4154517