Evertse, Jan-Hendrik On sums of S-units and linear recurrences. (English) Zbl 0547.10008 Compos. Math. 53, 225-244 (1984). This paper deals with some remarkable consequences of Schlickewei’s p- adic version of the method of Thue-Siegel-Roth-Schmidt. The most striking results deal with recurrent sequences \(u_ 0,u_ 1,u_ 2,..\). given by a linear recurrence relation \(u_{n+1}=a_ ku_ n+...+a_ 0u_{n- k}\), \(a_ i\) fixed. Here we assume \(a_ i,u_ n\in {\mathbb{Q}}\). Let P(a/b) be the greatest prime factor occurring in the numerator and denominator of a/\(b\in {\mathbb{Q}}\). One of the results is, that \(P(u_ r/u_ s)\to\infty \) if \(r>s\) and \(r\to\infty \). Some equally interesting results, including the ones that were obtained independently by A. J. van der Poorten, are also given. Reviewer: F.Beukers Cited in 21 ReviewsCited in 91 Documents MSC: 11B37 Recurrences 11D61 Exponential Diophantine equations 11J81 Transcendence (general theory) Keywords:linear recurrence relation; greatest prime factor PDFBibTeX XMLCite \textit{J.-H. Evertse}, Compos. Math. 53, 225--244 (1984; Zbl 0547.10008) Full Text: Numdam EuDML References: [1] S. Chowla : Proof of a conjecture of Julia Robinson . Norske Vid. Selsk. Forh.(Trondheim) 34 (1961) 107-109. · Zbl 0105.02803 [2] E. Dubois and G. Rhin , Sur la majoration de formes linéaires a coefficients algébriques réels et p-adiques. Démonstration d’une conjecture de K. Mahler . C.R. Acad. Sc. Paris 282, Série A-1211 (1976). · Zbl 0339.10031 [3] K. Gyory , On the number of solutions of linear equations in units of an algebraic number field . Comment. Math. Helv. 54 (1979) 583-600. · Zbl 0437.12004 [4] S. Lang , Integral points on curves . Inst. Hautes Etudes Sci. Publ. Math. no. 6 (1960) 27-43. · Zbl 0112.13402 [5] D.J. Lewis and K. Mahler , On the representation of integers by binary forms . Acta Arith. 6 (1961) 333-363. · Zbl 0102.03601 [6] K. Mahler , Zur Approximation algebraischer Zahlen (I). Uber den gröszten Primteiler Binärer Formen . Math. Ann. 107 (1933) 691-730. · Zbl 0006.10502 [7] K. Mahler : Math. Rev. 42 (1971) 3028. [8] T. Nagell , Sur une propriété des unités d’un corps algébrique . Arkiv för Mat. 5 (1965) 343-356. · Zbl 0128.03403 [9] T. Nagell : Quelques problèmes relatifs aux unités algébriques . Arkiv för Mat. 8 (1969) 115-127. · Zbl 0213.06902 [10] T. Nagell : Sur un type particulier d’unités algébriques . Arkiv för Mat. 8 (1969) 163-184. · Zbl 0213.06901 [11] M. Newman : Units in arithmetic progression in an algebraic number field . Proc. Amer. Math. Soc. 43 (1974) 266-268. · Zbl 0285.12011 [12] G. Polya , Arithmetische Eigenschaften der Reihenentwicklungen . J. reine angew. Math. 151 (1921) 1-31. · JFM 47.0276.02 [13] A.J. Van Der Poorten , Some problems of recurrent interest. Macquarie Math. Reports ; Macquarie Univ., Northride, Australia, 81-0037 (1981). [14] A.J. Van Der Poorten and H.P. Schlickewei , The growth conditions for recurrence sequences. Macquarie Math. Reports 82 -0041 (1982). [15] H.P. Schlickewei Über die diophantische Gleichung x 1+x2+...+xn=0 Acta Arith. 33 (1977) 183-185. · Zbl 0355.10017 [16] H.P. Schlickewei : The p-adic Thue-Siegel-Roth-Schmidt Theorem . Arch Math. 29 (1977) 267-270. · Zbl 0365.10026 [17] W.M. Schmidt : Simultaneous approximation to algebraic numbers by elements of a number field Monatsh. Math. 79 (1975) 55-66. · Zbl 0317.10042 [18] W.M. Schmidt : Diophantine Approximation , Lecture Notes in Math. 785, Springer Verlag, Berlin Etc. 1980. · Zbl 0421.10019 [19] Th. Schneider : Anwendung eines abgeänderten Roth-Ridoutschen Satzes auf diophantische Gleichungen . Math. Ann. 169 (1967) 177-182. · Zbl 0146.26904 [20] T.N. Shorey , Linear forms in numbers of a binary recursive sequence . Acta Arith, to appear. · Zbl 0491.10011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.