Partition regularity and the primes

摘要

The purpose of this Note is to point out, as a simple yet nice consequence of Green and Tao's program on counting linear patterns in the primes and Deuber's work on partition regularity, that if a system of equations is partition regular over the positive integers, then it is also partition regular over the sets {. p-. 1.:. p prime} as well as {. p+. 1.:. p prime}. This answers a question of Li and Pan.