An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)

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. So we have g(2) = 6 and g(3) = g(4) = 7. In this paper we present the following new results on the Erdős-Selfridge function g(k):(1) We present a new algorithm to compute the value of g(k), and use it to both verify previous work [1,16,12] and compute new values of g(k), with our current limit being g(323) = 1 69829 77104 46041 21145 63251 22499.(2) We define a new function ĝ(k), and under the assumption of our uniform distribution heuristic we show that