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

Christian Ballot

Acta Arithmetica (1999)

  • Volume: 89, Issue: 3, page 259-277
  • ISSN: 0065-1036

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Christian Ballot. "Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*." Acta Arithmetica 89.3 (1999): 259-277. <http://eudml.org/doc/207269>.

@article{ChristianBallot1999,
author = {Christian Ballot},
journal = {Acta Arithmetica},
keywords = {prime divisors; linear recurrence sequences; third order recurrences; Williams-Ballot sequences; Laxton-Ballot group; densities of maximal divisors},
language = {eng},
number = {3},
pages = {259-277},
title = {Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*},
url = {http://eudml.org/doc/207269},
volume = {89},
year = {1999},
}

TY - JOUR
AU - Christian Ballot
TI - Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*
JO - Acta Arithmetica
PY - 1999
VL - 89
IS - 3
SP - 259
EP - 277
LA - eng
KW - prime divisors; linear recurrence sequences; third order recurrences; Williams-Ballot sequences; Laxton-Ballot group; densities of maximal divisors
UR - http://eudml.org/doc/207269
ER -

References

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  2. [Ba2] C. Ballot, Group structure and maximal division for cubic recursions with a double root, Pacific J. Math. 173 (1996), 337-355. Zbl0866.11007
  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. Zbl0979.11007
  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. Zbl0139.27501
  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. Zbl0569.10003
  6. J. C. Lagarias, Sets of primes determined by systems of polynomial congruences, Illinois J. Math. 27 (1983), 224-237. Zbl0507.12004
  7. [Lax] R. R. Laxton, On groups of linear recurrences I, Duke Math. J. 26 (1969), 721-736. Zbl0226.10010
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  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. Zbl22.0220.01
  12. [Mo] P. Moree, On the prime density of Lucas sequences, J. Théor. Nombres Bordeaux 8 (1996), 449-459. Zbl0873.11058
  13. [M-S] P. Moree and P. Stevenhagen, Prime divisors of Lucas sequences, Acta Arith. 82 (1997), 403-410. Zbl0913.11048
  14. [Na] T. Nagell, Über die Einheiten in reinen kubischen Zahlkörpern, Skr. Norske Vid. Akad. Oslo Mat.-Naturv. Klasse 11 (1923), 1-34. Zbl49.0111.03
  15. [Wa1] M. Ward, The maximal prime divisors of linear recurrences, Canad. J. Math. 6 (1954), 455-462. Zbl0056.04106
  16. [Wa2] M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math. 11 (1961), 379-386. Zbl0112.26904
  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. Zbl0004.24402
  18. [Wi1] H. C. Williams, A generalization of the Lucas functions, unpublished Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1969. 
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  20. [Wi3] H. C. Williams, Edouard Lucas and Primality Testing, Canad. Math. Soc. Ser. Monographs Adv. Texts, Wiley, 1998. 

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