On additive partitions of integers

The aim of this paper is to prove that given a linear recurrence sequence U = kb u„ +z = u„ + , + u„, n w 1, u t =1, uz > u l , then the set of positive integers can be partitioned uniquely into two disjoint subsets such that the sum of any two distinct members from any one set can never be in U and study related problems . We prove our main result in Section 2. In Section 3 we give a graph theoretic interpretation of this, and look at such recurrence sequences as extremal solutions to certain problems relating to the partition of the set of integers . In Section 4 we make a brief study of some special properties of partitions generated by such recursive sequences . Finally in Section 5 we mention related problems and possible generalizations of our results . The theorem mentioned in the first paragraph has been proved simultaneously and independently, [2, 4, 6], by Evans, Silverman and Nelson for the case u z =2, (Fibonacci Numbers) . But we bave learned that their methods are quite different . Moreover in this paper we study the same problem in a more general setting . The explicit theorem originated by Silverman is [7] :