On Rainbow Solutions to an Equation with a Quadratic Term

In this paper, we prove that every 3-coloring of the positive integers such that the upper density of each color is greater than 4 contains a rainbow solution to a−b = c 2. A solution is rainbow if all of its elements are of different colors. Furthermore, the 4 bound is sharp. We also prove two results for rainbow solutions of a− b = c2 in Zn. One stipulates that if Zn, for an odd n, is partitioned into three color classes R,B,G with min {|R| , |B| , |G|} > n r1 , where r1 is the smallest prime factor of n, then there must always exist a rainbow solution to a− b ≡ c2 mod n. Our second theorem in Zn extends this, demonstrating that if we have min {|R| , |B| , |G|} > n 2r1 , then there exists a rainbow solution to a− b ≡ c2 mod n except in a very specific case, which we classify.