engleski

Diophantine quadruples in Z[i][X]

A set {;a_1, ..., a_m}; of m positive integers is called a Diophantine m-tuple, if the product of any of its two distinct elements increased by 1 is a perfect square. One of the obvious and most interesting questions is how large those sets can be. Very recently, the folklore conjecture, that there does not exist a Diophantine quintuple, was solved by He, Togbe and Ziegler. There is also a stronger version of that conjecture, which is still open, that states that every triple can be extended to quadruple with a larger element in a unique way. In this talk we will consider one of the generalization of this problem, considering such sets in the ring Z[i][X] instead of Z. More precisely, we prove that if {;a, b, c, d}; is a set of four non-zero polynomials from Z[i][X], not all constant, such that the product of any two of its distinct elements increased by1is a square of a polynomial from Z[i][X], then (a+b−c−d)^2=4(ab+1)(cd+1). This result obviously implies that we cannot have 5 polynomials form Z[i][X] with above mentioned property.

Diophantine m-tuples

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