engleski

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

In this paper, we study D(-1)-triples 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 and twice prime squared. We prove that in those cases c has to be an integer. 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 t in {; ; 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

This paper is accepted for publication on February 17, 2014. The final publication is available at link.springer.com.