On additive partitions of sets of positive integers

Abstract

Consider any set U = u n with elements defined by u n +2 = u n +2 + u n , n ⩾ 1, where u 1 and u 2 are relatively prime positive integers. We show that if u 1 < u 2 or 2¦ u 1 u 2 , then the set of positive integers can be partitioned uniquely into two disjoint sets such that the sum of any two distinct members of any one set is never in U . If u 1 > u 2 and 2 ∤ u 1 u 2 , no such partition is possible. Further related results are proved which generalize theorems of Alladi, Erdös, and Hoggatt.