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\begin{center}
\vskip 1cm{\LARGE\bf Binomial Coefficient Predictors
}
\vskip 1cm
\large
Vladimir Shevelev\\
Department of Mathematics \\
Ben-Gurion University of the Negev\\
Beer-Sheva 84105 \\
Israel \\
\href{mailto:shevelev@bgu.ac.il}{\tt shevelev@bgu.ac.il}\\
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\begin{abstract}
For a prime $p$ and nonnegative integers $n,k,$ consider the set
$A_{n, k}^{(p)}=\{x\in [0,1,...,n]: p^k||\binom {n} {x}\}.$ 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
{\sl binomial coefficient predictor in base $p$} ($p$-BCP),
if $|A_{n, k}^{(p)}|=\alpha_{k}p^{\nu-k},
k=0,1, \ldots, \nu.$ We give a full description of the $p$-BCP's in
every base $p.$
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