engleski

On the extension of the Diophantine pair {; ; 1, 3}; ; in Z[sqrt(d)]

A Diophantine m-tuple in a commutative ring R with the unit 1 is the set of m distinct non-zero elements with the property that the product of each two distinct elements increased by 1 is a perfect square in R. The most famous examples are quadruples {; ; 1/16, 33/16, 17/4, 105/16}; ; (found by Diophant) and {; ; 1, 3, 8, 120}; ; (found by Fermat). One of the most interesting problems here is the bound for the size of such Diophantine sets. In the ring of integers Z, it is known that there is no Diophantine 6-tuple and the conjecture says that there is no Diophantine quintuple. In the ring Z[sqrt(d)], no bound for the size of these sets is known. Here, we investigate the extension of the Diophantine pair {; ; 1, 3}; ; in Z[sqrt(d)] for d < 0 and that can be considered as small step towards determining an absolute bound for such sets in Z[sqrt(d)].

diophantine m-tuples

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