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 greater-than-or-equal-to 2 . The conjecture that p less-than-o r-equal-to max(N/k, 29) is supported by considerable numerical evidenc e. Call a binomial coefficient good if p > k For 1 less-than-or-equal i less-than-or-equal-to k write N - k + i = a(i)b(i) , where b(i) cont ains 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 ) > ck2/ln k is proved. We conjecture that our list of 17 binomial coe fficients with deficiency > 1 is complete, and it seems that the numbe r with deficiency 1 is finite. All (k(N)) with positive deficiency and k less-than-or-equal-to 101 are listed.