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Bartz-Marlewski equation with generalized Lucas components

Hayder R. Hashim (2022)

Archivum Mathematicum

Let { U n } = { U n ( P , Q ) } and { V n } = { V n ( P , Q ) } be the Lucas sequences of the first and second kind respectively at the parameters P 1 and Q { - 1 , 1 } . In this paper, we provide a technique for characterizing the solutions of the so-called Bartz-Marlewski equation x 2 - 3 x y + y 2 + x = 0 , where ( x , y ) = ( U i , U j ) or ( V i , V j ) with i , j 1 . Then, the procedure of this technique is applied to completely resolve this equation with certain values of such parameters.

Best simultaneous diophantine approximations of some cubic algebraic numbers

Nicolas Chevallier (2002)

Journal de théorie des nombres de Bordeaux

Let α be a real algebraic number of degree 3 over whose conjugates are not real. There exists an unit ζ of the ring of integer of K = ( α ) for which it is possible to describe the set of all best approximation vectors of θ = ( ζ , ζ 2 ) .’

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