On sets of integers whose shifted products are powers

摘要

Let N be a positive integer and let A be a subset of {1,...,N} with the property that aa' + 1 is a pure power whenever a and a' are distinct elements of A. We prove that vertical bar A vertical bar, the cardinality of A, is not large. In particular, we show that vertical bar A vertical bar (log N)(2/3) (log log N)(1/3). (C) 2007 Elsevier Inc. All rights reserved.