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Binomial Coefficient Predictors Vladimir Shevelev
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105
Israel
mailto:shevelev@bgu.ac.ilshevelev@bgu.ac.il

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Abstract:

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 $p.$