  Here we answer a conjecture by Ron Graham about getting finer upper bounds
for van der Waerden numbers in the affirmative, but without the application of
double induction or combinatorics as applied to sets of integers that contain
some van der Waerden number as an element. Rather we obtain the result solely
by exploiting certain properties of any integer greater than one that is
divisible by another integer. Our mathematical methods are easily accessible by
those whose field of specialization lies outside of combinatorial number
theory, such as discrete mathematics, elementary number theory or analytic
number theory.
