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Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

Let F n be the n -th Fibonacci number. Put ϕ = 1 + 5 2 . We prove that the following inequalities hold for any real α :1) inf n | | F n α | | ϕ - 1 ϕ + 2 ,2) lim inf n | | F n α | | 1 5 ,3) lim inf n | | ϕ n α | | 1 5 .These results are the best possible.

Diophantine equations with Euler polynomials

Dijana Kreso, Csaba Rakaczki (2013)

Acta Arithmetica

We determine decomposition properties of Euler polynomials and using a strong result relating polynomial decomposition and diophantine equations in two separated variables, we characterize those g(x) ∈ ℚ [x] for which the diophantine equation - 1 k + 2 k - + ( - 1 ) x x k = g ( y ) with k ≥ 7 may have infinitely many integer solutions. Apart from the exceptional cases we list explicitly, the equation has only finitely many integer solutions.

Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier (2005)

Journal de Théorie des Nombres de Bordeaux

We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " d -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. d -th root) of two (resp. one) linear recurrences implies that this quotient (resp. d -th root) is itself a recurrence. We shall also relate such...

Diophantine triples with values in binary recurrences

Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper, we study triples a , b and c of distinct positive integers such that a b + 1 , a c + 1 and b c + 1 are all three members of the same binary recurrence sequence.

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