engleski

On the extensibility of D(-1)-triples {;1, b, c}; in the ring Z[√ -t], t > 0

Let b = 2, 5, 10 or 17 and t > 0. We study the existence of D(-1)-quadruples of the form {;1, b, c, d}; in the ring Z[√ -t]. We prove that if {;1, b, c}; is a D(-1)-triple in Z[√ -t], then c is an integer. As a consequence of this result, we show that for t \not\in {;1, 4, 9, 16}; there does not exist a subset of Z[√ -t] of the form {;1, b, c, d}; with the property that the product of any two of its distinct elements diminished by 1 is a square of an element in Z[√ -t].

Diophantine quadruples; quadratic field; simultaneous Pellian equations; linear form in logarithms

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