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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.

Palindromic powers.

Hernández, Santos Hernández, Luca, Florian (2006)

Revista Colombiana de Matemáticas

Perfect powers in the summatory function of the power tower

Florian Luca, Diego Marques (2010)

Journal de Théorie des Nombres de Bordeaux

Let ( a n ) n 1 be the sequence given by a 1 = 1 and a n = n a n - 1 for n 2 . In this paper, we show that the only solution of the equation a 1 + + a n = m l is in positive integers l > 1 , m and n is m = n = 1 .

Primitive divisors of Lucas and Lehmer sequences, II

Paul M. Voutier (1996)

Journal de théorie des nombres de Bordeaux

Let α and β are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all α and β with h ( β / α ) 4 , the n -th element of these sequences has a primitive divisor for n > 30 . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

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