LINEAR EQUATIONS IN THE STONE-•CECH COMPACTIFICATION OF N

Let a and b be distinct positive integers. We show that the equation u + a ¢ p = v + b ¢ p has no solutions with u; v 2 flN and p 2 flNnN. More generally, we show that if (S;+) is any commutative cancellative semigroup and S has no nontrivial solutions to n ¢ s = n ¢ t for n 2 N and s; t 2 S, then the equation u + a ¢ p = v + b ¢ p has no solutions with u; v 2 flS and p 2 flSnS. We characterize completely the Abelian groups for which such an equation can be satisfled. We also show that if S can be embedded in the circle group T, then the equation a ¢ p + u = b ¢ p + v has no solutions with u; v 2 flS and p 2 flSnS. Finally, we investigate solutions to the equation a1 ¢ p+ a2 ¢ p+: ::+ an ¢ p = b1 ¢ p+ b2 ¢ p+: ::+ bm ¢ p where p 2 flNnN and a1; a2; : : : ; an; b1; b2; : : : ; bm 2 N. 1