Shifted products that are coprime pure powers

摘要

A set A of positive integers is called a coprime Diophantine powerset if the shifted product ab + 1 of two different elements a and b of A is always a pure power, and the occurring pure powers are all coprime. We prove that each coprime Diophantine powerset A subset of {1,..., N} has vertical bar A vertical bar <= 8000 log N/ log log N for sufficiently large N. The proof combines results from extremal graph theory with number theory. Assuming the famous abc-conjecture, we are able to both drop the coprimality condition and reduce the upper bound to c log log N for a fixed constant C. (c) 2004 Elsevier lnc. All rights reserved.