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Padovan and Perrin numbers as products of two generalized Lucas numbers

Kouèssi Norbert Adédji, Japhet Odjoumani, Alain Togbé (2023)

Archivum Mathematicum

Let P m and E m be the m -th Padovan and Perrin numbers respectively. Let r , s be non-zero integers with r 1 and s { - 1 , 1 } , let { U n } n 0 be the generalized Lucas sequence given by U n + 2 = r U n + 1 + s U n , with U 0 = 0 and U 1 = 1 . In this paper, we give effective bounds for the solutions of the following Diophantine equations P m = U n U k and E m = U n U k , where m , n and k are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

Padua and Pisa are exponentially far apart.

Benjamin M. M. De Weger (1997)

Publicacions Matemàtiques

We answer the question posed by Ian Stewart which Padovan numbers are at the same time Fibonacci numbers. We give a result on the difference between Padovan and Fibonacci numbers, and on the growth of Padovan numbers with negative indices.

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