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 $p(\binom{g(k)}{k}) >k$. 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 \[ g(360) = 4\ 73246\ 49994\ 16835\ 77114\ 34474\ 66861. \]

We prove that our algorithm runs in time sublinear in $g(k)$, and under the assumption of a reasonable heuristic, its running time is \[ g(k) \exp[ -c (k\log\log k) /(\log k)^2 (1+o(1))] \text{ for } c >0. \]

number theory

Audience: researchers in the topic

( chat | paper | slides | video )

Comments: Chairs: Alina Ostafe and Kate Stange

---

Algorithmic Number Theory Symposium (ANTS XIV)

Series comments: Registration is now open. Registration is free but required to access the chat and livestream.

This is a hybrid synchronous/asynchronous conference with several ways to participate.

• Click the "paper" link to view contributed papers and posters (open to all).
• Click the "video" link to view pre-recorded talks of accepted papers (open to all).
  These are 15-20 minutes aimed at a general algorithmic number theory audience.
• Click the "slides" link to view slides used in the pre-recorded video when available (open to all).
• Click the "chat" link to access the chat stream related to the talk or poster before, during, and after the live event (registration required).
• Click the "livestream" button to join the live event when it is taking place (registration required and you must be logged in).
  For accepted papers the audience will be expected to have watched the pre-recorded video and have the paper in front of them.

The invited talks will be recorded and made available via the "video" link after the talk is over. None of the other sessions will be recorded.

---