Primitive divisors of Lucas and Lehmer sequences
HTML articles powered by AMS MathViewer
- by Paul M. Voutier PDF
- Math. Comp. 64 (1995), 869-888 Request permission
Abstract:
Stewart reduced the problem of determining all Lucas and Lehmer sequences whose nth element does not have a primitive divisor to solving certain Thue equations. Using the method of Tzanakis and de Weger for solving Thue equations, we determine such sequences for $n \leq 30$. Further computations lead us to conjecture that, for $n > 30$, the nth element of such sequences always has a primitive divisor.References
- Geo. D. Birkhoff and H. S. Vandiver, On the integral divisors of $a^n-b^n$, Ann. of Math. (2) 5 (1904), no. 4, 173–180. MR 1503541, DOI 10.2307/2007263
- A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62. MR 1234835, DOI 10.1515/crll.1993.442.19
- A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
- R. D. Carmichael, On the numerical factors of the arithmetic forms $\alpha ^n\pm \beta ^n$, Ann. of Math. (2) 15 (1913/14), no. 1-4, 30–48. MR 1502458, DOI 10.2307/1967797
- J. H. E. Cohn, Squares in some recurrent sequences, Pacific J. Math. 41 (1972), 631–646. MR 316367
- B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325–367. MR 901244, DOI 10.1016/0022-314X(87)90088-6
- B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. MR 1026936
- B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR 0160744
- L. K. Durst, Exceptional real Lehmer sequences, Pacific J. Math. 9 (1959), 437–441. MR 108465
- Gregory Karpilovsky, Field theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 120, Marcel Dekker, Inc., New York, 1988. Classical foundations and multiplicative groups. MR 972982 W. Ljunggren, Über die unbestimmte Gleichung $A{x^2} - B{y^4} = C$, Arch. Math. Naturv. XLI, 10 (1938). —, Einige Sätze über unbestimmte Gleichungen von der Form $A{x^4} + B{x^2} + C = D{y^2}$, Skrifter utgitt av Det Norske Videnskaps-Akademi I Oslo I. Mat.-Naturv. Klasse. 9 (1942).
- W. Ljunggren, Some remarks on the diophantine equations $x^{2}-{\cal D}y^{4}=1$ and $x^{4}-{\cal D}y^{2}=1$, J. London Math. Soc. 41 (1966), 542–544. MR 197390, DOI 10.1112/jlms/s1-41.1.542
- Michael Pohst and Hans Zassenhaus, On effective computation of fundamental units. I, Math. Comp. 38 (1982), no. 157, 275–291. MR 637307, DOI 10.1090/S0025-5718-1982-0637307-6
- Neville Robbins, On Fibonacci numbers of the form $px^{2}$, where $p$ is prime, Fibonacci Quart. 21 (1983), no. 4, 266–271. MR 723787
- A. Schinzel, Primitive divisors of the expression $A^{n}-B^{n}$ in algebraic number fields, J. Reine Angew. Math. 268(269) (1974), 27–33. MR 344221, DOI 10.1515/crll.1974.268-269.27
- C. L. Stewart, Primitive divisors of Lucas and Lehmer numbers, Transcendence theory: advances and applications (Proc. Conf., Univ. Cambridge, Cambridge, 1976) Academic Press, London, 1977, pp. 79–92. MR 0476628
- C. L. Stewart, On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers, Proc. London Math. Soc. (3) 35 (1977), no. 3, 425–447. MR 491445, DOI 10.1112/plms/s3-35.3.425
- N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), no. 2, 99–132. MR 987566, DOI 10.1016/0022-314X(89)90014-0
- F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693–707. MR 669662, DOI 10.1090/S0025-5718-1982-0669662-5
- Morgan Ward, The intrinsic divisors of Lehmer numbers, Ann. of Math. (2) 62 (1955), 230–236. MR 71446, DOI 10.2307/1969677
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 869-888
- MSC: Primary 11D61; Secondary 11B39, 11Y50
- DOI: https://doi.org/10.1090/S0025-5718-1995-1284673-6
- MathSciNet review: 1284673