engleski

Finiteness results for F-Diophantine sets

Diophantine sets, i.e. sets of positive integers A with the property that the product of any two distinct elements of A increased by 1 is a perfect square, have a vast literature, dating back to Diophantus of Alexandria. The most important result states that such sets A can have at most ve elements, and there are only nitely many of them with ve elements. Beside this, there are a large number of niteness results, concerning the original problem and some of its many variants. In this paper we introduce the notion of, and prove niteness results on so called (F ; m)-Diophantine sets A, where F is a bivariate polynomial with integer coecients, and instead of requiring ab + 1 to be a square for all distinct a ; b in A, the numbers F(a ; b) should be full m-th powers. The particular choice F(x ; y) = xy + 1 and m = 2 gives back the original problem.

Diophantine sets; polynomials in two variables; binary forms; power values of polynomials

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