Distinct Consecutive Products

Abstract

We prove that there is a density-one set of positive integers for which the products of all distinct consecutive blocks are distinct, answering a question of Erdős and Graham. The construction retains or rejects entire interiors of consecutive-prime gaps and is therefore stable under passage to longer prefixes. Uniform affine-curve estimates control the witnesses with no previous rejected gap, and a scale-contracting forest controls every remaining witness unconditionally.