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 aa′+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|≪(logN)2/3(loglogN)1/3.