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

We study D(-1)-triple of the form {;1, b, c}; in the ring Z[√-t], t > 0, for positive integer b such that b is a prime, twice prime or twice prime squared. We prove that in those cases c has to be an integer. As a consequence of this result, in cases of b = 26, 37 or 50 we prove that D(−1)-triples of the form {;1, b, c}; cannot be extended to a D(−1)-quadruple in the ring Z[√-t], t > 0, except in cases of t ∈ {;1, 4, 9, 25, 36, 49};. For those exceptional cases of t we show that there exist infinitely many D(−1)-quadruples of the form {;1, b, −c, d};, c, d > 0 in Z[√−t].

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

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