Divisibility of central binomial coefficients

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge1$, let $B_n=\binom{2n}{n}=(2n)!/(n!)^2$ and let $\gcd$ denote the greatest common divisor of positive integers.

Statement recorded in the supplied source.

Settle both assertions:

1. The set $\{n\ge1:\gcd(B_n,3\cdot5\cdot7)=1\}$ is infinite.
2. The set $\{n\ge1:\gcd(B_n,3\cdot5\cdot7\cdot11)=1\}$ is finite.

For an odd prime $p$, the condition $p\nmid B_n$ is equivalently that every digit of $n$ in base $p$ is at most $(p-1)/2$. [S2]

Short English statement

Determine whether avoidance of the primes three, five and seven occurs infinitely often for central binomial coefficients, while simultaneous avoidance of eleven occurs only finitely often.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3383 (OPG-506). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Divisibility of central binomial coefficients, OPG-506. Original problem statement, full mathematical discussion, bibliography and comments. https://www.openproblemgarden.org/op/divisibility_of_central_binomial_coefficients.