Binomial coefficient predictors

Shevelev, Vladimir

Journal of integer sequences, Tome 14 (2011) no. 2.

Résumé

Summary: For a prime $ p$ and nonnegative integers $ n,k,$ consider the set $ A_{n, k}^{(p)}=\{x\in [0,1,...,n]: p^k\vert\vert\binom {n} {x}\}.$ Let the expansion of $ n+1$ in base $ p$ be $ n+1=\alpha_{0} p^{\nu}+\alpha_{1}p^{\nu-1}+\cdots+\alpha_{\nu},$ where $ 0\leq \alpha_{i}\leq p-1, i=0, \ldots, \nu.$ Then $ n$ is called a binomial coefficient predictor in base $ p ( p$-BCP), if $ \vert A_{n, k}^{(p)}\vert=\alpha_{k}p^{\nu-k}, k=0,1, \ldots, \nu.$ We give a full description of the $ p$-BCP's in every base $ p.$

Détail

Classification : 11B65, 11A07, 11A15
Keywords: binomial coefficient, maximal exponent of a prime dividing an integer, p-ary expansion of integer, Kummer's theorem