How to find the least upper bound on the van der Waerden Number $W(r, k)$ that is some integer Power of the coloring Integer $r$

What is a least integer upper bound on van der Waerden number W (r, k) among the powers of the integer r? We show how this can be found by expanding the integer W (r, k) into powers of r. Doing this enables us to find both a least upper bound and a greatest lower bound on W (r, k) that are some powers of r and where the greatest lower bound is equal to or smaller than W (r, k). A finite series expansion of each W (r, k) into integer powers of r then helps us to find also a greatest real lower bound on any k for which a conjecture posed by R. Graham is true, following immediately as a particular case of the overall result.12