Van Der Waerden Numbers (vdwnumbers.org) is a research project that aims to find better lower bounds for these numbers. The sequence of colors BRRBBRRB (where B is blue and R is red) does not have an evenly spaced subsequence of length 3 that are the same color. However, if you add a B to the end, you get BRRBBRRBB, which has the same color blue in positions 1,5, and 9 which are evenly spaced 4 apart. If you add an R to the end, you get BRRBBRRBR, which has R at position 3, 6, and 9. In fact, with only two colors, there is no sequence of length 9 of Bs and Rs that does not have a subsequence of 3 evenly spaced of the same color. Van der Waerden's Theorem states that for any number of colors r and length k, a long enough sequence always has an evenly spaced subsequence of length k of the same color. The smallest length guaranteed to have an evenly spaced subsequence is called the Van Der Waerden Number and is written W(k,r). For example, W(3,2)=9. Only 7 of these numbers are known. This project is to find better lower bounds for Van Der Waerden Numbers by finding sequences like BRRBBRRB using large prime numbers with special properties.

This project is now launched with URL: 123numbers.org.