Erdős #684 · number theory / primes

A classical theorem of Mahler states that for any $\epsilon>0$ and integers $k$ and $l$ then, writing

where the only primes dividing $a$ are $\leq l$ and the only primes dividing $b$ are $>l$, we have $a < n^{1+\epsilon}$ for all sufficiently large (depending on $\epsilon,k,l$) $n$.

Mahler's theorem implies $f(n)\to \infty$ as $n\to \infty$, but is ineffective, and so gives no bounds on the growth of $f(n)$.

One can similarly ask for estimates on the smallest integer $f(n,k)$ such that if $m$ is the factor of $\binom{n}{k}$ containing all primes $\leq f(n,k)$ then $m > n^2$.

Tang and ChatGPT have proved that

The same proof would prove $f(n) \leq n^{2/3+o(1)}$ assuming the Riemann Hypothesis (or Density Hypothesis).

A heuristic given by Sothanaphan and ChatGPT in the comments suggests that, at least for most $n$, $f(n)\sim 2\log n$.

An internal OpenAI model [APSSV26] has given an elementary argument which shows $f(n) \ll (\log n)^2$, and constructs arbitrarily large $n$ such that $f(n)\geq (1/2-o(1))\log n$.

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