Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-30T22:11:10.307Z Has data issue: false hasContentIssue false
  • English
  • Français

On the growth of recurrence sequences

Published online by Cambridge University Press:  24 October 2008

J. H. Loxton  and
A. J. van der Poorten
J. H. Loxton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W., Australia, 2033
A. J. van der Poorten
Affiliation:
School of Mathematics, University of New South Wales, Kensington, N.S.W., Australia, 2033
Get access Rights & Permissions [Opens in a new window]

Extract

In this note, we discuss some questions on the arithmetic properties of recurrence sequences. Our primary purpose is to mention some methods which originated as lemmas in certain transcendence studies, but in order to give context to these ideas, we briefly review other techniques applicable to the problems considered. These problems concern the distribution of zeros in a recurrence, the rate of growth of its terms and the size of the greatest prime factor of the terms.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 3 , May 1977 , pp. 369 - 376
Copyright
Copyright © Cambridge Philosophical Society 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Cassels, J. W. S.An embedding theorem for fields. Bull. Austral. Math. Soc. 14 (1976), 193–198.CrossRefGoogle Scholar
Cassels, J. W. S.Addendum: Bull. Austral. Math. Soc. 14 (1976), 479–480.CrossRefGoogle Scholar
(2)Kubota, K. K. On a conjecture of Morgan Ward: I, II. (To appear.)Google Scholar
(3)Laxton, R. R.Linear recurrences of order 2. J. Austral. Math. Soc. 7 (1967), 108–114.CrossRefGoogle Scholar
(4)Lech, C.A note on recurring series. Ark. Mat. 2 (1953), 417–421.CrossRefGoogle Scholar
(5)Lewis, D. J. Diophantine equations: p-adic methods. In ‘Studien in number theory’, ed. LeVeque, W. J.. Studies in Mathematics 6 (MAA, 1969), 25–75.Google Scholar
(6)Loxton, J. H. and Van Der Poorten, A. J.Arithmetic properties of certain functions in several variables. J. Number Theory (to appear).Google Scholar
(7)Mahler, K.Über das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen. Math. Ann. 103 (1930), 573–587.CrossRefGoogle Scholar
(8)Mahler, K.Eine arithmetische Eigenschaft der rekurrierenden Reihe. Mathematica (Leiden) 3 (1934/1935), 153–156.Google Scholar
(9)Mahler, K.Eine arithmetische Eigenschaft der Taylor Koeffizienten rationaler Funktionen. Proc. Akad. Wet. Amsterdam 38 (1935), 51–60.Google Scholar
(10)Mahler, K.On the Taylor coefficients of rational functions. Proc. Cambridge Philos. Soc. 52 (1956), 39–48.CrossRefGoogle Scholar
Mahler, K.Addendum: Proc. Cambridge Philos. Soc. 53 (1957), 544.CrossRefGoogle Scholar
(11)Mahler, K.A remark on recursive sequences. J. Math. Sci. 1 (1966), 12–17.Google Scholar
(12)Mignotte, M. Suites récurrentes linéaires. Sém. Delange-Pisot-Poitou, 15e année (1973/1934), no. G 14.Google Scholar
(13)Mignotte, M.A note on linear recursive sequences. J. Austral. Math. Soc. (A) 20 (1975), 242–244.CrossRefGoogle Scholar
(14)Poorten, A. J. Van Der. Generalisations of Turdn's main theorems on lower bounds for sums of powers. Bull. Austral. Math. Soc. 2 (1970), 15–37.CrossRefGoogle Scholar
(15)Poorten, A. J. Van Der. Zeros of p-adic exponential polynomials. Nederl. Akad. Wetensch Indag. Math. 38 (1976), 46–49.CrossRefGoogle Scholar
(16)Poorten, A. J. Van Der. Hermite interpolation and p-adic exponential polynomials. J. Austral. Math. Soc. 22 (1976), 12–26.CrossRefGoogle Scholar
(17)Poorten, A. J. Van Der and Tijdeman, R.On common zeros of exponential polynomials. Enseignement Math. 21 (1975), 57–67.Google Scholar
(18)Ritt, J. F.A factorisation theory for functions. Trans. Amer. Math. Soc. 29 (1927), 584–596.Google Scholar
(19)Schinzel, A.On two theorems of Gelfond and some of their applications. Acta Arith. 13 (1967), 177–236.CrossRefGoogle Scholar
Schinzel, A.Corrigendum: Acta Arith. 16 (1969/1970), 101.CrossRefGoogle Scholar
(20)Skolem, T. Ein Verfahren zur Behandlung gewisser exponentialer Gleichungen und dio phantischer Gleichungen. Comptes rendus du 8e Congrès des Mathematicians scandinaves (Stockholm, 1934), pp. 163–188. (Lund, Häkan Ohlssons Boktryckeri, 1935.)Google Scholar
(21)Stewart, C. L. Divisor properties of arithmetical sequences. Ph.D. thesis, University of Cambridge (1976).Google Scholar
(22)Stewart, C. L. On the divisors of Fermat, Fibonacci, Lucas and Lehmer numbers. (To appear.)Google Scholar
(23)Strassman, R.Über den Wertevorrat von Potenzreihen im Gebiet der p-adischen Zahlen. J. Reine Angew. Math. 159 (1928), 13–28.CrossRefGoogle Scholar
(24)Ward, M.Some diophantine problems connected with linear recurrences. Report of the Institute in the Theory of Numbers (University of Colorado, 1959), pp. 250–257.Google Scholar