engleski

On the family of Diophantine pairs {;P_{;2k};,P_{;2k+2};};

Let k≥1 be an integer and let P_k be the k-th Pell number and Q_k k-th Pell-Lucas number. In this paper, we prove that if d is a positive integer such that {;P_{;2k};, P_{;2k+2};, 2P_{;2k+2};, d}; is a Diophantine quadruple, then d=P_{;2k+1};Q_{;2k+1};Q_{;2k+2};. We deduce that the pair {;P_{;2k};, 2P_{;2k+2};}; cannot be extended to an irregular Diophantine quadruple.

Diophantine m-tuples ; system of Pellian equations ; reduction method ; linear formin logarithms

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