This paper provides new lower bounds for van der Waerden numbers. The number
W(k,r) is defined to be the smallest integer n for which any r-coloring
of the integers 0β¦,nβ1 admits monochromatic arithmetic progression of
length k; its existence is implied by van der Waerden's Theorem. We exhibit
r-colorings of 0β¦nβ1 that do not contain monochromatic arithmetic
progressions of length k to prove that W(k,r)>n. These colorings are
constructed using existing techniques. Rabung's method, given a prime p and a
primitive root Ο, applies a color given by the discrete logarithm base
Ο mod r and concatenates kβ1 copies. We also used Herwig et al's
Cyclic Zipper Method, which doubles or quadruples the length of a coloring,
with the faster check of Rabung and Lotts. We were able to check larger primes
than previous results, employing around 2 teraflops of computing power for 12
months through distributed computing by over 500 volunteers. This allowed us to
check all primes through 950 million, compared to 10 million by Rabung and
Lotts. Our lower bounds appear to grow roughly exponentially in k. Given that
these constructions produce tight lower bounds for known van der Waerden
numbers, this data suggests that exact van der Waerden Numbers grow
exponentially in k with ratio r asymptotically, which is a new conjecture,
according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader
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