A note on perfect powers of the form $x^{m-1} + ... + x + 1$

Mao-hua, LE

doi:10.4064/aa-69-1-91-98

Cited by 7 publications

(2 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We warn the reader that the results obtained by [8] and [18] remain unproved. Indeed, they all depend on lemma 3 of [8], which is incorrect (see the comment of Yuan Ping-Zhi [19]). 2, where the above-mentioned corollaire 3 of [6] allows the authors to bound n independently of a.…”

Section: Applications To the Diophantine Equation X N −1mentioning

confidence: 91%

Linear forms in p-adic logarithms and the Diophantine equation formula here

Bugeaud

1999

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

A longstanding conjecture claims that the Diophantine equationformula herehas finitely many solutions and, maybe, only those given byformula hereAmong the known results, let us mention that Ljunggren [9] solved (1) completely when q = 2 and that very recently Bugeaud et al. [3] showed that (1) has no solution when x is a square. For more information and in particular for finiteness type results under some extra hypotheses, we refer the reader to Nagell [11], Shorey & Tijdeman [16, 17] and to the recent survey of Shorey [15]. One of the main tools used in most of the proofs is Baker's theory of linear form in archimedean logarithms of algebraic numbers and especially a dramatic sharpening obtained when the algebraic numbers involved are closed to 1. This was first noticed by Shorey [14], and has been applied in numerous works relating to (1) or to the Diophantine equationformula heresee for instance [1, 5, 10]. The purpose of the present work is to prove a similar sharpening for linear forms in p-adic logarithms and to show how it can be applied in the context of (1). Further, following previous investigations by Sander [12] and Saradha & Shorey [13], we derive from our results an irrationality statement for Mahler's numbers.

show abstract

Section: Applications To the Diophantine Equation X N −1mentioning

confidence: 91%

Linear forms in p-adic logarithms and the Diophantine equation formula here

Bugeaud

1999

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

show abstract

“…The ®rst of these results was stated as a corollary in Le [Le2], but the proof is erroneous since Lemma 3 of [Le2] is false (this also invalidates the claims of Yu and Le in [LY]; see the comment in [Yu]). Similarly, the results of Le [Le3] on equation (1.5) must be regarded as unproven since Lemma 1 and Lemma 2 of that paper are both incorrect; indeed the fact that the equation X 2 À 3Y 2 11 2 possesses integral solutions while X 2 À 3Y 2 11 is insoluble serves to contradict Lemma 1 while, in the notation of [Le3], taking aY bY nY kY X Y Y 1Y 2Y 3Y 47Y 63Y 50 contradicts Lemma 2.…”

Section: Introductionmentioning

confidence: 99%

Rational approximation to algebraic numbers of small height: the Diophantine equation |axn - byn|= 1

Bennett¹

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

136

View full text Add to dashboard Cite

Following an approach originally due to Mahler and sharpened by Chudnovsky, we develop an explicit version of the multi-dimensional``hypergeometric method'' for rational and algebraic approximation to algebraic numbers. Consequently, if aY b and n are given positive integers with n 3, we show that the equation of the title possesses at most one solution in positive integers xY y. Further results on Diophantine equations are also presented. The proofs are based upon explicit Pade Â approximations to systems of binomial functions, together with new Chebyshev-like estimates for primes in arithmetic progressions and a variety of computational techniques.F xY y m

show abstract