An algorithm and estimates for the Erdős–Selfridge function

Abstract

Let $p(n)$ denote the smallest prime divisor of the integer $n$. Define the function $g(k)$ to be the smallest integer $>k+1$ such that $p(\binom{g(k)}{k})>k$. So we have $g(2)=6$ and $g(3)=g(4)=7$. In this paper we present the following new results on the Erd\H{o}s-Selfridge function $g(k)$: We present a new algorithm to compute the value of $g(k)$, and use it to both verify previous work and compute new values of $g(k)$, with our current limit being $$ g(323)= 1\ 69829\ 77104\ 46041\ 21145\ 63251\ 22499. $$ We define a new function $\hat{g}(k)$, and under the assumption of our Uniform Distribution Heuristic we show that $$ \log g(k) = \log \hat{g}(k) + O(\log k) $$ with high "probability". We also provide computational evidence to support our claim that $\hat{g}(k)$ estimates $g(k)$ reasonably well in practice. There are several open conjectures on the behavior of $g(k)$ which we are able to prove for $\hat{g}(k)$, namely that $$ 0.525\ldots +o(1) \quad \le \quad \frac{\log \hat{g}(k)}{k/\log k} \quad \le \quad 1+o(1), $$ and that $$ \limsup_{k\rightarrow\infty} \frac{\hat{g}(k+1)}{\hat{g}(k)}=\infty.$$ Let $G(x,k)$ count the number of integers $n\le x$ such that $p(\binom{n}{k})>k$. Unconditionally, we prove that for large $x$, $G(x,k)$ is asymptotic to $x/\hat{g}(k)$. And finally, we show that the running time of our new algorithm is at most $g(k) \exp[ -c (k\log\log k) /(\log k)^2 (1+o(1))]$ for a constant $c>0$.