Shifted products that are coprime pure powers

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 ⊂ {1,....., N} has |A| ≤ 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.

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