Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Hamed Mousavi
Affiliation:

Georgia Tech
Date:

Wed, 12/01/2022 - 14:30 - 15:30 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 [3])

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/246

[3] mailto:moree@mpim-bonn.mpg.de