Post #3682663

AI-assisted answers to Erdős conjectures get all the buzz, but good old-fashioned human mathematicians [*] recently settled a 50-year-old conjecture: Is there a positive constant c such that every binomial coefficient Bin(n,k) has a divisor in the interval (cn,n]? The answer turns out to be: YES when n is large and k is large compared to n, but strictly NO. It's a long proof, but I managed to learn something cool from the first few pages. Erdős originally guessed that maybe Bin(n,k) would always have a divisor in the interval [n-k+1,n], which would be a tighter result. After all: Bin(n,k) = n (n-1) (n-2 ) ... (n-k+1) / k! and it's easy to imagine k! failing to “spoil” *all* of the factors in the numerator here. But in 1958, Schinzel found the counterexample Bin(99215, 15), which is NOT divisible by any of 99201 ... 99215. In fact, Schinzel found an infinite number of counterexamples of the form Bin(n, 15), where the values of n form an arithmetic series, and for n = 13085213870159810495 this is also a counterexample to the conjecture that [cn,n] might always contain a divisor for the choice c=3/4. So this was an early hint that maybe even the conjecture allowing for any choice of c would fail ... but it took until 2026 to settle this! https://arxiv.org/abs/2605.21221v2 * Edit: the authors did make some use of AI tools, but it seems to have been predominantly a collaboration between between several authors combining their own results on aspects of the proof.