Estimates of the least prime factor of a binomial coefficient

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.