### On a conjecture of Graham concerning the p-divisibility of central binomial coefficients

Combinatorics Seminar

24th January 2023, 11:00 am – 12:00 pm

Fry Building, 2.04

In this talk, we study a question of Graham. Are there infinitely many integers n for

which the central binomial coefficient {2n \choose n} is relatively prime to 105 = 3x5x7? By elementary number theory, this is the same as asking if there are infinitely many integers n, such that n added to itself base 3, 5, or 7, has no carries, and probabilistic heuristics suggest there should be infinitely many such integers n. We establish a statistical version of this result and its generalization for an arbitrary number of primes, \{p_1, . . . , p_N\}:

There are infinitely many n such that n+n has “few” carries mod p_i, for each i.

Our method lives at the interface of Fourier analysis, analytic number theory, and the

geometry of numbers. This is joint work with Ernie Croot and Maxie Schmidt (Georgia

Tech).

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