On sets of integers whose shifted products are powers

Let N be a positive integer and let A be a subset of { 1 , … , N } with the property that a a ′ + 1 is a pure power whenever a and a ′ are distinct elements of A. We prove that | A | , the cardinality of A, is not large. In particular, we show that | A | ≪ ( log N ) 2 / 3 ( log log N ) 1 / 3 .