First occurrences of square-free gaps and an algorithm for their computation

Abstract

This paper reports the results of a search for first occurrences of square-free gaps using an algorithm based on the sieve of Eratosthenes. Using Qgap(L) to denote the starting number of the first gap having exactly the length L, the following values were found since August 1999: Qgap(10)=262315467, Qgap(12)=47255689915, Qgap(13)=82462576220, Qgap(14)=1043460553364, Qgap(15)=79180770078548, Qgap(16)=3215226335143218, Qgap(17)=23742453640900972 and Qgap(18)=125781000834058568. No gaps longer than 18 were found up to N=125870000000000000.