engleski

On the extensibility of parametric family of Diophantine triples for Gaussian integers

We consider the parametric family of Diophantine triples {;k-1, k+1, 4k}; in the ring of Gaussian integers. It has the property that the product of its any two distinct elements increased by 1 is a perfect square. We prove that if {;k-1, k+1, 4k, d};, for k in Z[i] \ {;0, +-};, is a Diophantine quadruple, i.e. if (k-1)d+1, (k+1)d+1 and 4kd+1 are perfect squares in Z[i], then d=16k^3-4k.

Diophantine quadruples; Simultaneous Diophantine

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