Binomial Coefficient Predictors

Vladimir Shevelev Department of Mathematics Ben-Gurion University of the Negev Beer-Sheva 84105 Israel

                  shevelev@bgu.ac.il

Abstract

For a prime p and nonnegative integers n, k, consider the set A_{n,k}^{(p)} = {x ∈ [0, 1, ..., n] : p^{k}||\binom{n}{x}\}. Let the expansion of n + 1 in base p be n + 1 = α_{0}p^{ν} + α_{1}p^{ν−1} + ⋯ + α_{ν}, where 0 ≤ α_{i} ≤ p − 1, i = 0, …, ν. Then n is called a binomial coefficient predictor in base p(p-BCP), if |A_{n,k}^{(p)}| = α_{k}p^{ν−k}, k = 0, 1, …, ν. We give a full description of the p-BCP’s in every base p.