New Lower Bounds for van der Waerden Numbers Using Distributed Computing

TL;DR

This study establishes new lower bounds for van der Waerden numbers by leveraging distributed computing to analyze a vast range of primes, providing evidence for their exponential growth with respect to colors and progression length.

Contribution

It introduces a large-scale distributed computing approach to improve lower bounds for van der Waerden numbers, extending the range of primes analyzed significantly beyond prior work.

Findings

New lower bounds for van der Waerden numbers up to primes of 950 million.

Evidence supporting that van der Waerden numbers grow roughly as r^k.

Distributed computing enables extensive prime analysis for combinatorial number theory.

Abstract

This paper provides new lower bounds for van der Waerden numbers using Rabung's method, which colors based on the discrete logarithm modulo some prime. Through a distributed computing project with 500 volunteers over one year, we checked all primes up to 950 million, compared to 27 million in previous work. We point to evidence that the van der Waerden number for $r$ colors and progression length $k$ is roughly $r^k$.