engleski

A polynomial variant of a problem of Diophantus and its consequences

In this paper we prove that every Diophantine quadruple in R[X] is regular. In other words, we prove that if {; ; ; ; a, b, c, d}; ; ; ; is a set of four non-zero elements of R[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 R[X], then (a+b−c−d)^2=4(ab+1)(cd+1). Some consequences of the above result are that for an arbitrary n ∈N there does not exist a set of five non-zero elements from Z[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 Z[X]. Furthermore, there can exist such a set of four non-zero elements of Z[X] if and only if n is a square.

Diophantine m-tuples ; polynomials

nije evidentirano