On the Existence of Perfect Splitter Sets

03/01/2019

∙

by   Pingzhi Yuan, et al.

∙

0

∙

share

Given integers k_1, k_2 with 0< k_1<k_2, the determinations of all positive integers q for which there exists a perfect Splitter B[-k_1, k_2](q) set is a wide open question in general. In this paper, we obtain new necessary and sufficient conditions for an odd prime p such that there exists a nonsingular perfect B[-1,3](p) set. We also give some necessary conditions for the existence of purely singular perfect splitter sets. In particular, we determine all perfect B[-k_1, k_2](2^n) sets for any positive integers k_1,k_2 with k_1+k_2>4. We also prove that there are infinitely many prime p such that there exists a perfect B[-1,3](p) set.