Let and are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all and with h, the -th element of these sequences has a primitive divisor for . In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.
Soit et deux entiers algébriques complexes conjugués. On propose un algorithme dont l’objet est de découvrir des éléments des suites de Lucas ou de Lehmer associées à et , n’ayant pas de diviseurs primitifs. On utilise cet algorithme pour démontrer que pour tout et tel que , le -ième terme des suites de Lucas et de Lehmer admet un diviseur primitif dès que . Nous donnons en outre une amélioration d’un résultat de Stewart se rapportant à des suites plus générales.
@article{JTNB_1996__8_2_251_0, author = {Paul M. Voutier}, title = {Primitive divisors of {Lucas} and {Lehmer} sequences, {II}}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {251--274}, publisher = {Universit\'e Bordeaux I}, volume = {8}, number = {2}, year = {1996}, zbl = {0873.11013}, mrnumber = {1438469}, language = {en}, url = {https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_251_0/} }
TY - JOUR AU - Paul M. Voutier TI - Primitive divisors of Lucas and Lehmer sequences, II JO - Journal de théorie des nombres de Bordeaux PY - 1996 SP - 251 EP - 274 VL - 8 IS - 2 PB - Université Bordeaux I UR - https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_251_0/ LA - en ID - JTNB_1996__8_2_251_0 ER -
Paul M. Voutier. Primitive divisors of Lucas and Lehmer sequences, II. Journal de théorie des nombres de Bordeaux, Volume 8 (1996) no. 2, pp. 251-274. https://jtnb.centre-mersenne.org/item/JTNB_1996__8_2_251_0/
[1] On the size of the coefficients of the cyclotomic polynomial, Topics in Classical Number Theory,, (Budapest, 1981), Colloquia Mathematica Societatis Janos Bolyai, 34, North-Holland, New York, 1984. | MR | Zbl
, and ,,[2] On the integral divisors of an - bn, Ann. of Math. (2) 5 (1904), 173-180. | JFM | MR
and ,[3] On the coefficients of the cyclotomic polynomials, Amer. Math. Monthly 75 (1968), 372-377. | MR | Zbl
,[4] On the numerical factors of the arithmetic forms an ±βn, Ann. of Math. (2) 15 (1913), 30-70. | JFM
,[5] Exceptional real Lehmer sequences, Pacific J. Math. 9 (1959), 437-441. | MR | Zbl
[6] An Introduction to the Theory of Numbers, Oxford University Press, 5th edition, 1978. | MR | Zbl
and ,[7] Field Theory: Classical Foundations and Multiplicative Groups, Marcel Dekker, New York, 1988. | MR | Zbl
,[8] Formes Linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory, to appear. | Zbl
, and ,[9] The distribution of totatives, Canadian J. Math. 7 (1955), 347-357. | MR | Zbl
,[10] Lower bounds for linear forms in logarithms, New Advances in Transcendence Theory (A. Baker, ed.), Cambridge University Press, Cambridge, 1988. | MR
and ,[11] Estimation de la fonction de Tchebychef θ sur le k-ième nombre premier et grandes valeurs de la fonction w(n) nombre de diviseurs premiers de n, Acta Arith. XLII (1983), 367-389. | EuDML | Zbl
,[12] Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64-94. | MR | Zbl
and ,[13] Primitive divisors of the expression An - Bn in algebraic number fields, J. Reine Angew. Math. 268/269 (1974), 27-33. | EuDML | MR | Zbl
,[14] Primitive divisors of Lucas and Lehmer sequences, Transcendence Theory: Advances and Applications (A. Baker and D.W. Masser, eds.), Academic Press, New York, 1977. | MR
,[15] On divisors of Fermat, Fibonacci, Lucas and Lehmer numbers, Proc. London Math. Soc. (3) 35 (1977), 425-447. | MR | Zbl
,[16] Primitive divisors of Lucas and Lehmer sequences, Math. Comp. 64 (1995), 869-888. | MR | Zbl
,[17] An effective lower bound for the height of algebraic numbers, Acta Arith., (to appear). | MR | Zbl
,[18] Linear Independence of Logarithms of Algebraic Numbers, IMSc Report No 116 (1992), The Institute of Mathematical Sciences, Madras. | Zbl
,[19] The intrinsic divisors of Lehmer numbers, Ann. of Math. (2) 62 (1955), 230-236. | MR | Zbl
,[20] Zur Theorie der Potenzreste, Monatsh. Math. 3 (1892), 265-284. | JFM | MR
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