## Primitive divisors of Lucas and Lehmer sequences.(English)Zbl 0832.11009

Let $$\alpha$$, $$\beta$$ be algebraic numbers such that $$\alpha+ \beta$$ and $$\alpha \beta$$ are relatively prime rational integers and $$\alpha/ \beta$$ is not a root of unity. The sequence $$(u_n )_{n\geq 0}$$ defined by $$u_n= (\alpha^n- \beta^n)/ (\alpha- \beta)$$ is called a Lucas sequence. If we replace the condition $$\alpha+ \beta\in \mathbb{Z}$$ by $$(\alpha+ \beta)^2\in \mathbb{Z}$$ then, another sequence of rational integers can be defined by $$u_n= (\alpha^n- \beta^n)/ (\alpha- \beta)$$ if $$n$$ is odd and $$u_n= (\alpha^n- \beta^n)/ (\alpha^2- \beta^2)$$ if $$n$$ is even, which is called a Lehmer sequence. A prime number $$p$$ is a primitive divisor of a Lucas (resp. Lehmer) number $$u_n$$ if $$p\mid u_n \wedge p\nmid (\alpha- \beta)^2 u_2 \dots u_{n-1}$$ (resp. $$p\nmid (\alpha^2- \beta^2)^2 u_3\dots u_{n-1})$$.
C. L. Stewart has proved that any Lucas (resp. Lehmer) number $$u_n$$ has a primitive divisor, provided that $$n> e^{452} 2^{67}$$ (resp. $$n> e^{452} 4^{67}$$). He also proved that for any $$n= 5, 7, 8, 9, 10, 11, \dots$$ (resp. $$n= 7, 9, 11, 13, 14, 15, \dots$$) there are at most finitely many Lucas (resp. Lehmer) numbers $$u_n$$.
The author of the present paper explicitly determines all Lucas numbers $$u_n$$ for $$n= 5, 7, 8, 9, 10, \dots, 30$$ (such numbers actually occur only for $$n= 5, 7, 8, 10, 12, 13, 18$$ and 30) and all Lehmer numbers $$u_n$$ for $$n= 7, 9, 11, 13, 14, 15, \dots, 30$$ (these actually occur only for $$n= 7, 9, 13, 14, 15, 18, 24, 26$$ and 30). He reduces the problem of their determination to the explicit solution of diophantine equations (integer solutions) of type $$aX^2- bY^4 =c$$ ($$a, b, c$$ pairwise relatively prime integers with $$a$$, $$b$$ positive) in the cases $$n= 5, 8, 10$$ and three instances of case $$n= 12$$, and of Thue type for all other values of $$n$$. These solutions of the non-trivial cases $$aX^2- bY^4 =c$$ are due to Ljunggren, Cohn or Robbins, while the Thue equations (their degrees vary from 3 to 14) are solved by the general method of N. Tzanakis and B. M. M. de Weger [J. Number Theory 31, 99–132 (1989; Zbl 0657.10014)].
The paper is very readable, as the author gives sufficient technical details, which help the reader for a deeper understanding of the method. It is worth noticing that, to the best of the reviewer’s knowledge, this is the first paper in the literature in which Thue equations of this degree are solved.

### MSC:

 11B37 Recurrences 11D41 Higher degree equations; Fermat’s equation 11Y50 Computer solution of Diophantine equations

Zbl 0657.10014
Full Text:

### References:

 [1] Geo. D. Birkhoff and H. S. Vandiver, On the integral divisors of \?$$^{n}$$-\?$$^{n}$$, Ann. of Math. (2) 5 (1904), no. 4, 173 – 180. · JFM 35.0205.01 · doi:10.2307/2007263 [2] A. Baker and G. Wüstholz, Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19 – 62. · Zbl 0788.11026 · doi:10.1515/crll.1993.442.19 [3] A. I. Borevich and I. R. Shafarevich, Number theory, Translated from the Russian by Newcomb Greenleaf. Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. · Zbl 0145.04902 [4] R. D. Carmichael, On the numerical factors of the arithmetic forms \?$$^{n}$$\pm \?$$^{n}$$, Ann. of Math. (2) 15 (1913/14), no. 1-4, 30 – 48. · JFM 44.0216.01 · doi:10.2307/1967797 [5] J. H. E. Cohn, Squares in some recurrent sequences, Pacific J. Math. 41 (1972), 631 – 646. · Zbl 0248.10016 [6] B. M. M. de Weger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987), no. 3, 325 – 367. · Zbl 0625.10013 · doi:10.1016/0022-314X(87)90088-6 [7] B. M. M. de Weger, Algorithms for Diophantine equations, CWI Tract, vol. 65, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. · Zbl 0687.10013 [8] 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. · Zbl 0133.30202 [9] L. K. Durst, Exceptional real Lehmer sequences, Pacific J. Math. 9 (1959), 437 – 441. · Zbl 0091.04204 [10] 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. · Zbl 0677.12010 [11] W. Ljunggren, Über die unbestimmte Gleichung $$A{x^2} - B{y^4} = C$$, Arch. Math. Naturv. XLI, 10 (1938). · JFM 64.0975.05 [12] -, 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). [13] W. Ljunggren, Some remarks on the diophantine equations \?²-\cal\?\?$$^{4}$$=1 and \?$$^{4}$$-\cal\?\?²=1, J. London Math. Soc. 41 (1966), 542 – 544. · Zbl 0147.02505 · doi:10.1112/jlms/s1-41.1.542 [14] Michael Pohst and Hans Zassenhaus, On effective computation of fundamental units. I, Math. Comp. 38 (1982), no. 157, 275 – 291. , https://doi.org/10.1090/S0025-5718-1982-0637307-6 Michael Pohst, Peter Weiler, and Hans Zassenhaus, On effective computation of fundamental units. II, Math. Comp. 38 (1982), no. 157, 293 – 329. · Zbl 0493.12004 [15] Neville Robbins, On Fibonacci numbers of the form \?\?², where \? is prime, Fibonacci Quart. 21 (1983), no. 4, 266 – 271. · Zbl 0523.10003 [16] A. Schinzel, Primitive divisors of the expression \?$$^{n}$$-\?$$^{n}$$ in algebraic number fields, J. Reine Angew. Math. 268/269 (1974), 27 – 33. Collection of articles dedicated to Helmut Hasse on his seventy-fifth birthday, II. · Zbl 0287.12014 · doi:10.1515/crll.1974.268-269.27 [17] 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. [18] C. L. Stewart, On divisors of Fermat, Fibonacci, Lucas, and Lehmer numbers, Proc. London Math. Soc. (3) 35 (1977), no. 3, 425 – 447. · Zbl 0389.10014 · doi:10.1112/plms/s3-35.3.425 [19] 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. · Zbl 0657.10014 · doi:10.1016/0022-314X(89)90014-0 [20] F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693 – 707. · Zbl 0505.12010 [21] Morgan Ward, The intrinsic divisors of Lehmer numbers, Ann. of Math. (2) 62 (1955), 230 – 236. · Zbl 0065.27102 · doi:10.2307/1969677 [22] Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. · Zbl 0484.12001
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.