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On Lucas pseudoprimes of the form \(ax^2+bxy+cy^2\) in arithmetic progression \(AX+B\) with a prescribed value of the Jacobi symbol. (English) Zbl 1028.11008
Let \(F\) be a binary indefinite integral quadratic form and assume it has odd fundamental discriminant \(d\) and belongs to the principal genus. Consider the sequence of Lucas numbers with parameters \(P\) and \(Q\) and \(D= P^2-4Q\). Then, under some natural conditions on \(P,Q,D,A,B\), if the arithmetic progression \(AX+B\) represents at all an odd integer \(n_0\) with the Jacobi symbol \((D/n_0)=\varepsilon\), where \(\varepsilon = \pm 1\) is fixed, then the progression represents infinitely many Lucas pseudoprimes \(n\) with parameters \(P,Q\), represented by the quadratic form \(F\) and satisfying \((D/n)=\varepsilon\).
MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A15 Power residues, reciprocity
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