A Schur-type Theorem for Primes

Thus if all primes are colored with k colors, then there exist arbitrarily long monochromatic arithmetic progressions. This is a van der Waerden-type [9] theorem for primes. (The well-known van der Waerden theorem states that for any m-coloring of all positive integers, there exist arbitrarily long monochromatic arithmetic progressions.) On the other hand, Schur’s theorem [7] is another important result in the Ramsey theory for integers. Schur’s theorem asserts that for any m-coloring of all positive integers, there exist monochromatic x, y, z such that x+ y = z. In this paper, we shall prove a Schur-type theorem for primes.