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!
arXiv.orgBinomial coefficients with divisors avoiding an intervalWe solve a fifty-year-old conjecture of Erdős and Graham concerning whether the binomial coefficient ${n \choose k}$ with $1 \leq k \leq \frac{n}{2}$ must always have a divisor $\leq n$ 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 $\leq n$ close to $n$. This latter, more substantial argument involves a restricted covering problem with residue classes, sieve methods, and various exponential sum estimates.