On a problem of Graham, Erdos, and Pomerance on the $p-$divisibility of central binomial coefficients

Erdos, Graham, Ruzsa, and Straus  proved that for primes $p,q$, there are infinitely many integers $n$ such that $\gcd\left(\binom{2n}{n},pq\right)=1$. Later Erdos and Graham conjectured that $\gcd\left(\binom{2n}{n},105\right)=1$ is true for infinitely many integers $n$. In this talk, we will prove a generalized statistical version of this conjecture. In particular, we prove that for $r>1$ and a set of  primes $p_1,\cdots, p_r$, there are infinitely many $n$, such that $\binom{2n}{n}$ is divisible by these primes with multiplicity of size at most $o(\log(n))$. This is equivalent to saying we can find integers $n$ whose base $p_1$, base $p_2$, $\cdots$, and base $p_r$ expansions all simultaneously have almost all their digits "small".  Our proof involves bypassing a deep, unsolved problem in Diophantine approximation and algebraic number theory, called Schanuel's Conjecture, through the use of a number of methods from analytic number theory and additive combinatorics.

This is joint work with Ernie Croot and Maxie Schmidt.
 

For zoom details please contact Pieter Moree (moree@mpim-bonn.mpg.de)