Diophantine Properties of Linear Recursive Sequences I

Pethö, Attila

doi:10.1007/978-94-011-5020-0_34

Cited by 11 publications

(11 citation statements)

References 36 publications

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“…• If n > 2 and F n = y p then p < 5.1 × 10 17 ; this was proved by Pethő using a linear form in three logarithms [27]. In the same paper he also showed that if n > 2 and L n = y p then p < 13222 using a linear form in two logarithms.…”

Section: A Brief Survey Of Previous Resultsmentioning

confidence: 89%

Fibonacci numbers at most one away from a perfect power

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

Elem. Math.

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Abstract. The famous problem of determining all perfect powers in the Fibonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ± 1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat's Last Theorem with Baker's Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary.

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Section: A Brief Survey Of Previous Resultsmentioning

confidence: 89%

Fibonacci numbers at most one away from a perfect power

Bugeaud¹,

Luca²,

Mignotte³

et al. 2008

Elem. Math.

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“…Very recently, Pethő [17] used the above result of Corvaja and Zannier to show that there are only finitely many perfect powers in a third order linear recurring sequence (G n ), if we assume that the characteristic polynomial of (G n ) is irreducible and has a dominating root.…”

Section: 1)mentioning

confidence: 99%

Polynomial-exponential equations and linear recurrences

Fuchs¹

2003

Glas. Mat. Ser. III

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Abstract. Let K be an algebraic number field and let (Gn) be a linear recurring sequence defined by Gn = λ 1 α n 1 +P 2 (n)α n 2 +· · ·+Pt(n)α n t , where λ 1 , α 1 , . . . , αt are non-zero elements of K and whereIn this paper we want to study the polynomial-exponential Diophantine equation f (Gn, x) = 0. We want to use a quantitative version of W. M. Schmidt's Subspace Theorem (due to J.-H. Evertse [8]) to calculate an upper bound for the number of solutions (n, x) under some additional assumptions.

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“…To be precise, if we apply a theorem of Mignotte [16], after reducing the problem to consideration of suitable Thue equations (see Section 2 of this paper), we obtain an upper bound for n in (1.1) of order log D. Similarly, a general result for powers in recurrence sequences due to Pethő [20] yields a bound for n in terms of the fundamental unit in Q( √ D). What is (debatably!)…”

Section: For Integers X and Q Larger Than One Then The Maximum Of Xmentioning

confidence: 81%

Powers in recurrence sequences: Pell equations

Bennett

2004

Trans. Amer. Math. Soc.

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Abstract. In this paper, we present a new technique for determining all perfect powers in so-called Pell sequences. To be precise, given a positive nonsquare integer D, we show how to (practically) solve Diophantine equations of the formin integers x, y and n ≥ 2. Our method relies upon Frey curves and corresponding Galois representations and eschews lower bounds for linear forms in logarithms. Along the way, we sharpen and generalize work of Cao, Af Ekenstam, Ljunggren and Tartakowsky on these and related questions.

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Diophantine Properties of Linear Recursive Sequences I

Cited by 11 publications

References 36 publications

Fibonacci numbers at most one away from a perfect power

Fibonacci numbers at most one away from a perfect power

Polynomial-exponential equations and linear recurrences

Powers in recurrence sequences: Pell equations

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