Estimates of the least prime factor of a binomial coefficient

We estimate the least prime factor p of the binomial coefficient ( {_k^N} ) for k ≥ 2 . The conjecture that p ≤ max (N/k,29) is supported by considerable numerical evidence. Call a binomial coefficient good if p > k . For 1 ≤ i ≤ 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 ( {_k^N} ) 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 ( {_k^N} ) with positive deficiency and k ≤ 101 are listed.