Japan Tobacco (Switzerland)

Unbounded logarithmic limsup in Erdős Problem 684 via shifted carry scheduling

For 1 ≤ k ≤ n, let u(n,k) be the part of the binomial coefficient binom(n,k) supported on primes at most k, and let f(n) be the least k for which u(n,k) > n^2. Erdős Problem 684 asks for bounds on f(n). We prove limsup_{n→∞} (f(n)/log n)(log log log n/log log n) ≥ 1/2. In particu...