Diophantine m-tuples for quadratic polynomials

Abstract

Abstract The poster presents one polynomial variant of the problem of Diophantus, described by A. Jurasic [Diophantine m-tuples for quadratic polynomials, Glas. Mat. Ser. III 46 (2011), 283–309], and ilustrates that results with some examples from the paper of A. Dujella and A. Jurasic [Some Diophantine triples and quadruples for quadratic polynomials, J. Comb. Number Theory 3(2) (2011), 123–141]. We proved that there does not exist a set with more than 98 nonzero polynomials in Z [ X ] , such that the product of any two of them plus a quadratic polynomial n is a square of a polynomial from Z [ X ] (we exclude the possibility that all elements of such set are constant multiples of a linear polynomial p ∈ Z [ X ] such that p 2 | n ). Specially, we prove that if such a set contains only polynomials of odd degree, then it has at most 18 elements.