%0 Journal Article
%T A polynomial variant of a problem of Diophantus and its consequences
%A Filipin
%A Alan
%A Jurasi£¿
%A Ana
%J -
%D 2019
%R 10.3336/gm.54.1.03
%X Sa£¿etak In this paper we prove that every Diophantine quadruple in £¿ [X] is regular. In other words, we prove that if {a, b, c, d} is a set of four non-zero elements of £¿[X], not all constant, such that the product of any two of its distinct elements increased by 1 is a square of an element of £¿[X], then (a+b-c-d)2=4(ab+1)(cd+1). Some consequences of the above result are that for an arbitrary n£¿ there does not exist a set of five non-zero elements from £¿[X], which are not all constant, such that the product of any two of its distinct elements increased by n is a square of an element of £¿[X]. Furthermore, there can exist such a set of four non-zero elements of £¿[X] if and only if n is a square
%K Diophantine m-tuples
%K polynomials
%U https://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=322403