2026/05/20 by Hung M. Bui, Slava Naprienko, Kyle Pratt +1 · 1 voice
Mathematics · #math.NT
We solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient n \choose k with 1 ≤ k ≤ (n)/(2) must always have a divisor ≤ n that is ``close'' to n: that is, bigger than a constant times n. We show this is the case when k is sufficiently large as a function of n. However, we show it is possible to find binomial coefficients n \choose k, where k is small compared to n, such that n \choose k does not have divisors ≤ n close to n. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.