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Estimates of the least prime factor of a binomial coefficient

Authors: P. Erdős, C. B. Lacampagne and J. L. Selfridge
Journal: Math. Comp. 61 (1993), 215-224
MSC: Primary 11B65; Secondary 11N37
MathSciNet review: 1199990
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Abstract: We estimate the least prime factor p of the binomial coefficient $ \left( {_k^N} \right)$ for $ k \geq 2$. The conjecture that $ p \leq \max (N/k,29)$ is supported by considerable numerical evidence. Call a binomial coefficient good if $ p > k$. For $ 1 \leq i \leq k$ write $ N - k + i = {a_i}{b_i}$, where $ {b_i}$ contains just those prime factors $ > k$ , and define the deficiency of a good binomial coefficient as the number of i for which $ {b_i} = 1$. Let $ g(k)$ be the least integer $ N > k + 1$ such that $ \left( {_k^N} \right)$ is good. The bound $ g(k) > c{k^2}/\ln k$ is proved. We conjecture that our list of 17 binomial coefficients with deficiency $ > 1$ is complete, and it seems that the number with deficiency 1 is finite. All $ \left( {_k^N} \right)$ with positive deficiency and $ k \leq 101$ are listed.

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Article copyright: © Copyright 1993 American Mathematical Society

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