Vol. 4, No. 1, 2020

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An algorithm and estimates for the Erdős–Selfridge function

Brianna Sorenson, Jonathan Sorenson and Jonathan Webster

Vol. 4 (2020), No. 1, 371–385
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(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(375) = 12863999653788432184381680413559.

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(kloglogk)(logk)2(1 + o(1))] for c > 0.

Keywords
Erdos–Selfridge function, elementary number theory, analytic number theory, binomial coefficients
Mathematical Subject Classification 2010
Primary: 11Y16
Supplementary material

Algorithm and estimates for the Erdos--Selfridge function

Milestones
Received: 20 February 2020
Revised: 7 September 2020
Accepted: 7 September 2020
Published: 29 December 2020
Authors
Brianna Sorenson
Mathematics, Statistics and Actuarial Science
Butler University
Indianapolis, IN
United States
Jonathan Sorenson
Computer Science and Software Engineering
Butler University
Indianapolis, IN
United States
Jonathan Webster
Mathematics, Statistics and Actuarial Science
Butler University
Indianapolis, IN
United States