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

TL;DR

This paper presents an algorithm based on the sieve of Eratosthenes to find the first occurrences of square-free gaps of various lengths, reporting specific starting points and the absence of longer gaps up to a large number.

Contribution

It introduces a new algorithm for efficiently computing the first occurrences of square-free gaps and provides extensive computational results.

Findings

First occurrences of gaps of lengths 10 to 18 identified.

No gaps longer than 18 found up to 125870000000000000.

Specific starting points for each gap length reported.

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.