Some binomial coefficient congruences
\textit{É. Lucas} [Sur les congruences des nombres Eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. Fr. 6, 49-54 (1878; JFM 10.0139.04)] has proved that if p is prime, \(n,r,n_ 0,r_ 0\) are nonnegative integers, \(n_ 0<p\), \(r_ 0<p\), then \(\left( \begin{matrix} np+n_ 0\\ rp+r_ 0\end{matrix} \right)\equiv \left( \begin{matrix} n\\ r\end{matrix} \right)\left( \begin{matrix} n_ 0\\ r_ 0\end{matrix} \right)(mod p)\). It is now proved that if p is prime, n,r,i are nonnegative integers, \(0<i<p\), then \[ \left( \begin{matrix} np\\ rp+i\end{matrix} \right)\equiv (r+1)\left( \begin{matrix} n\\ r+1\end{matrix} \right)\left( \begin{matrix} p\\ i\end{matrix} \right)(mod p^ 2). \] It is also shown that if \(p\geq 5\) is prime, \(i>0\), \(0\leq n\leq m\), \(0\leq k<p\), then \[ \left( \begin{matrix} mp^ 2\\ np^ 2+kp+i\end{matrix} \right)\equiv (n+1)\left( \begin{matrix} m\\ n+1\end{matrix} \right)\left( \begin{matrix} p^ 2\\ kp+i\end{matrix} \right)(mod p^ 3). \] The first proof proceeds by induction on n and the second by induction on m.
scientific article; zbMATH DE number 4210227