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Every positive integer greater than a positive integer $r$ can be written asan integer that is the sum of powers of $r$. Here we use this to prove theconjecture posed by Ronald Graham, B. Rothschild and Joel Spencer back in thenineteen nineties, that the van der Waerden number with $r$ colorings and witharithmetic progressions of $k$ terms, has a certain upper bound. Our proof doesnot need the application of double induction, constructive methods of proof orcombinatorics, as applied to sets of integers that contain some van der Waerdennumber 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 ofspecialization lies outside of combinatorial number theory, such as discretemathematics, computational complexity, elementary number theory or analyticnumber theory.

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 $W(r, k) < r^{k^{2}}$