16:00
An old conjecture of Graham asks whether there are infinitely many integers n such that \binom{2n}{n} is coprime to 105. This is equivalent to asking whether there are infinitely many integers which only have the digits 0,1 in base 3, 0,1,2 in base 5, and 0,1,2,3 in base 7. In general, one can ask whether there are infinitely many integers which only have 'small' digits in multiple bases simultaneously. For two bases this was established in 1975 by Erdos, Graham, Ruzsa, and Straus, but the case of three or more bases is much more mysterious. I will discuss recent joint work with Ernie Croot, in which we prove that (assuming the bases are sufficiently large) there are infinitely many integers such that almost all of the digits are small in all bases simultaneously.