Shifted products that are coprime pure powers (Mathematics Subject classification: Primary 11B75, 11D99; Secondary 05D10, 05C38)

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 occuring pure powers are all coprime. We prove that each coprime Diophantine powerset A ⊂ {1, . . . , N} has |A| ≤ 8000 logN/ log logN 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 logN for a fixed constant c.