Polynomial-exponential equations and linear recurrences

Fuchs, Clemens

doi:10.3336/gm.38.2.03

Cited by 15 publications

(29 citation statements)

References 23 publications

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“…This paper will not be concerned with quantitative aspects, though the methods allow to estimate the number of relevant solutions. In the context of the paper by Corvaja and Zannier ( [3]), some extimates have been obtained by Fuchs [7], using a quantitative version of the Subspace Theorem due to Evertse (see [6]). …”

Section: Introductionmentioning

confidence: 99%

Diophantine inequalities with power sums

Scremin¹

2007

J. Théor. Nombres Bordeaux

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Section: Introductionmentioning

confidence: 99%

Diophantine inequalities with power sums

Scremin¹

2007

J. Théor. Nombres Bordeaux

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“…In particular, t is uniquely determined, and is a perfect square. When (u n ) n≥0 = (F n ) n≥0 is the Fibonacci sequence, we have r = 1, so t = 1/4, which explains the example in (6).…”

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confidence: 96%

“…. , β t are algebraic numbers, but a close inspection of the proof of the main result in [6] shows that the β j can be chosen to be of the form α…”

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confidence: 99%

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Diophantine equations with products of consecutive terms in Lucas sequences II

Luca

Shorey

2008

Acta Arith.

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To Wolfgang Schmidt at his seventy-fifth birthday 1. Introduction. Let r and s be nonzero integers such that ∆ = r 2 + 4s = 0 and let α, β be the two roots of the quadratic equation x 2 −rx−s = 0. We assume further that α/β is not a root of 1. Let (u n ) n≥0 and (v n ) n≥0 be the Lucas sequences of the first and second kind with roots α and β, given by u n = (α n − β n )/(α − β) and v n = α n + β n for all n ≥ 0, respectively. These sequences can also be defined by u 0 = 0, u 1 = 1, v 0 = 2, v 1 = r and the recurrence relations u n+2 = ru n+1 + su n and v n+2 = rv n+1 + sv n for all n ≥ 0. When r = s = 1, the resulting sequences (u n ) n≥0 and (v n ) n≥0 are the sequences of Fibonacci numbers (F n ) n≥0 and Lucas numbers (L n ) n≥0 , and when r = 3, s = −2, the resulting sequence u n = 2 n − 1 for n ≥ 0 is the sequence of Mersenne numbers.In [7], we investigated Diophantine equations of the form

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“…The following important result is due to Corvaja and Zannier (see [13]). For extensions and generalizations of this result, see [14], [19] and [20].…”

Section: Proof Of Theoremmentioning

confidence: 99%

Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations

Bilu

Fuchs²,

Luca

2013

Publ. Math. Debrecen

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Abstract. We look at Diophantine equations arising from equating classical counting functions such as perfect powers, binomial coefficients and Stirling numbers of the first and second kind. The proofs of the finiteness statements that we give use a variety of methods from modern number theory, such as effective and ineffective tools from Diophantine approximation. As a tool for one part of the statements we establish a theoretical result that gives a more precise description on the structure of the solution set in the theorem, due to Bilu and Tichy, on Diophantine equations with separate variables in the case when infinitely many solutions exist.

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Polynomial-exponential equations and linear recurrences

Cited by 15 publications

References 23 publications

Diophantine inequalities with power sums

Diophantine inequalities with power sums

Diophantine equations with products of consecutive terms in Lucas sequences II

Combinatorial Diophantine equations and a refinement of a theorem on separated variables equations

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