WHEN SETS CAN OR CANNOT BE PRODUCT-DOMINANT

Abstract

Given a finite set [Formula presented], we define [Formula presented] A set A is said to be sum-dominant or MSTD (More Sums than Differences) if |A + A| > |A – A| and a set [Formula presented] is said to be product-dominant or MPTQ (More Products than Quotients) if |A.A| > |A/A|. In this paper, we shall discuss several properties of MPTQ sets, investigate techniques of generating an infinite family of MPTQ sets, and identify some characterizations under which a finit e set of numbers can or cannot be product-dominant. We confirm the existence of MPTQ sets of perfect squares and justify that nth powers of prime numbers do not contain any MPTQ set. We extend the notion of MPTQ sets to the multiplicative group [Formula presented] and recognize their correspondence with the MSTD sets in [Formula presented]. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.