The Expansion of each van der Waerden number $W(r, k)$ into Powers of $r$, when $r$ is the Number of Integer Colorings, determines a greatest lower Bound for all $k$ such that $W(r, k) < r^{k^{2}}$

TL;DR

This paper proves a conjecture about van der Waerden numbers, establishing a lower bound for all k where these numbers are less than r^{k^2}, using a novel approach based on properties of large integers.

Contribution

It provides a new proof of a conjecture on van der Waerden numbers without relying on combinatorial or inductive methods, using properties of large integers instead.

Findings

Established a lower bound for van der Waerden numbers in terms of powers of r.

Proved the conjecture by Graham, Rothschild, and Spencer from the 1990s.

The proof avoids complex combinatorial techniques, making it accessible to broader fields.

Abstract

Every positive integer greater than a positive integer $r$ can be written as an integer that is the sum of powers of $r$. Here we use this to prove the conjecture posed by Ronald Graham, B. Rothschild and Joel Spencer back in the nineteen nineties, that the van der Waerden number with $r$ colorings and with arithmetic progressions of $k$ terms, has a certain upper bound. Our proof does not need the application of double induction, constructive methods of proof or combinatorics, as applied to sets of integers that contain some van der Waerden number as an element. The proof instead derives from certain \emph{a priori} knowledge that is known about any positive integer when the integer is large. The mathematical methods we use are easily accessible by those whose field of specialization lies outside of combinatorial number theory, such as discrete mathematics, computational complexity, elementary number theory or analytic number theory.