engleski

Applications of a Diophantine equation of a special type

This is a joint work with Andrej Dujella and Mirela Jukic Bokun. Let p be an odd prime and k non-negative integer. We prove that the Pellian equation x^2 − (p^(2k+2) + 1)y^2 = −p^(2l+1), l ∈ {; ; ; 0, 1, . . . , k}; ; ; , is not solvable in positive integers x and y. By combining that result with other known results on the topic of Diophantine m-tuples, we obtain results on extensibility of D(−1)-pairs of the form {; ; ; 1, 2b}; ; ; , where 2b = p^(2j) + 1, j > 0, and p is an odd prime, to D(−1)-quadruples in the ring Z[√−t], t > 0.

Diophantine quadruples ; quadratic field ; Pellian equations ;

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