On the prime factorization of binomial coefficients

Abstract

Abstract For positive integers n and k , with n ≥2 k , let , where each prime factor of u is less than k , and each prime factor of v is at least equal to k . It is shown that u<v holds with just 12 exceptions, which are determined. If , where each prime factor of U is at most equal to k , and each prime factor of V is greater than k , then U < V holds with at most finitely many exceptions, 19 of which are determined. It is conjectured that there are no others.