#### 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 $p\left(n\right)$ denote the smallest prime divisor of the integer $n$. Define the function $g\left(k\right)$ to be the smallest integer $>k+1$ such that $p\left(\left(\genfrac{}{}{0.0pt}{}{g\left(k\right)}{k}\right)\right)>k$. We present a new algorithm to compute the value of $g\left(k\right)$, and use it to both verify previous work and compute new values of $g\left(k\right)$, with our current limit being

$g\left(375\right)=12863999653788432184381680413559.$

We prove that our algorithm runs in time sublinear in $g\left(k\right)$, and under the assumption of a reasonable heuristic, its running time is

##### Keywords
Erdos–Selfridge function, elementary number theory, analytic number theory, binomial coefficients
Primary: 11Y16
##### Supplementary material

Algorithm and estimates for the Erdos--Selfridge function