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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.

Palindromic powers.

Hernández, Santos Hernández, Luca, Florian (2006)

Revista Colombiana de Matemáticas

Parametric geometry of numbers and applications

Wolfgang M. Schmidt, Leonhard Summerer (2009)

Acta Arithmetica

Partial sums of $ζ (\frac{1}{2})$ modulo 1.

Vinson, Jade (2001)

Experimental Mathematics

Pell and Pell-Lucas numbers of the form $- 2^{a} - 3^{b} + 5^{c}$

Yunyun Qu, Jiwen Zeng (2020)

Czechoslovak Mathematical Journal

In this paper, we find all Pell and Pell-Lucas numbers written in the form $- 2^{a} - 3^{b} + 5^{c}$ , in nonnegative integers $a$ , $b$ , $c$ , with $0 \leq max {a, b} \leq c$ .

Perfect powers in products of integers from a block of consecutive integers

T. Shorey (1987)

Acta Arithmetica

Perfect powers in products of integers from a block of consecutive integers (II)

T. N. Shorey, Yu. V. Nesterenko (1996)

Acta Arithmetica

Perfect powers in the summatory function of the power tower

Florian Luca, Diego Marques (2010)

Journal de Théorie des Nombres de Bordeaux

Let ${(a_{n})}_{n \geq 1}$ be the sequence given by $a_{1} = 1$ and $a_{n} = n^{a_{n - 1}}$ for $n \geq 2$ . In this paper, we show that the only solution of the equation $a_{1} + \dots + a_{n} = m^{l}$ is in positive integers $l > 1, m$ and $n$ is $m = n = 1$ .

Périodes et isogénies des variétés abéliennes sur les corps de nombres

Jean-Benoît Bost (1994/1995)

Séminaire Bourbaki

Periodic Jacobi-Perron expansions associated with a unit

Brigitte Adam, Georges Rhin (2011)

Journal de Théorie des Nombres de Bordeaux

We prove that, for any unit $ϵ$ in a real number field $K$ of degree $n + 1$ , there exits only a finite number of n-tuples in $K^{n}$ which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for $n = 1$ . For $n = 2$ we give an explicit algorithm to compute all these pairs.

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