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Let $P_{m}$ and $E_{m}$ be the $m$ -th Padovan and Perrin numbers respectively. Let $r, s$ be non-zero integers with $r \geq 1$ and $s \in {- 1, 1}$ , let ${U_{n}}_{n \geq 0}$ 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.

Pair correlation of lattice points with prime constraint

Maosheng Xiong, Alexandru Zaharescu (2012)

Acta Arithmetica

Pairs of additive quadratic forms modulo one

R. C. Baker, S. Schäffer (1992)

Acta Arithmetica

Pairs of polynomials small at a number to certain algebraic powers

W. Dale Brownawell (1975/1976)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Palindromic continued fractions

Boris Adamczewski, Yann Bugeaud (2007)

Annales de l’institut Fourier

In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.

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