Download PDF - Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*All rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*

Authors 1

Affiliations

  1. Département de Mathématiques, Université de Caen, Campus 2, BP 5186, 14032 Caen Cedex, France

Bibliography

  1. [Ba1] C. Ballot, Density of prime divisors of linear recurrences, Mem. Amer. Math. Soc. 551 (1995).
  2. [Ba2] C. Ballot, Group structure and maximal division for cubic recursions with a double root, Pacific J. Math. 173 (1996), 337-355.
  3. [Ba3] C. Ballot, The density of primes p, such that -1 is a residue modulo p of two consecutive Fibonacci numbers, is 2/3, Rocky Mountain J. Math., to appear.
  4. [Ha] H. Hasse, Über die Dichte der Primzahlen p, für die eine vorgegebene ganzrationale Zahl a≠0 von gerader bzw. ungerader Ordnung mod p ist, Math. Ann. 166 (1966), 19-23.
  5. J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math. 118 (1985), 449-461; Errata: Pacific J. Math. 162 (1994), 393-396.
  6. J. C. Lagarias, Sets of primes determined by systems of polynomial congruences, Illinois J. Math. 27 (1983), 224-237.
  7. [Lax] R. R. Laxton, On groups of linear recurrences I, Duke Math. J. 26 (1969), 721-736.
  8. [Le1] D. H. Lehmer, On the multiple solutions of the Pell equation, Ann. of Math. (2) 30 (1929), 66-72.
  9. [Le2] D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, J. London Math. Soc. 10 (1935), 162-165.
  10. [Lu] E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math. 1 (1878), 184-240, 289-321.
  11. [Ma] G. B. Mathews, On the arithmetical theory of the form x³ + ny³ + n²z³ - 3nxyz, Proc. London Math. Soc. 21 (1890), 280-287.
  12. [Mo] P. Moree, On the prime density of Lucas sequences, J. Théor. Nombres Bordeaux 8 (1996), 449-459.
  13. [M-S] P. Moree and P. Stevenhagen, Prime divisors of Lucas sequences, Acta Arith. 82 (1997), 403-410.
  14. [Na] T. Nagell, Über die Einheiten in reinen kubischen Zahlkörpern, Skr. Norske Vid. Akad. Oslo Mat.-Naturv. Klasse 11 (1923), 1-34.
  15. [Wa1] M. Ward, The maximal prime divisors of linear recurrences, Canad. J. Math. 6 (1954), 455-462.
  16. [Wa2] M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math. 11 (1961), 379-386.
  17. [We] A. E. Western, On Lucas's and Pepin's tests for primeness of Mersenne numbers, J. London Math. Soc. 7 (1932), 130-137.
  18. [Wi1] H. C. Williams, A generalization of the Lucas functions, unpublished Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1969.
  19. [Wi2] H. C. Williams, On a generalization of the Lucas functions, Acta Arith. 20 (1972), 33-51.
  20. [Wi3] H. C. Williams, Edouard Lucas and Primality Testing, Canad. Math. Soc. Ser. Monographs Adv. Texts, Wiley, 1998.
Acta Arithmetica
1999/89/3
Export to PBN, Opens in new tab
Pages:
259-277
Main language of publication
English
Received
1998-09-10
Accepted
1998-12-01
Published
1999
Exact and natural sciences