Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*
Christian Ballot
Acta Arithmetica, Tome 89 (1999), p. 259-277 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)
Publié le : 1999-01-01
EUDML-ID : urn:eudml:doc:207269 
@article{bwmeta1.element.bwnjournal-article-aav89i3p259bwm,
     author = {Christian Ballot},
     title = {Strong arithmetic properties of the integral solutions of X3 + DY3 + D2Z3 - 3DXYZ = 1, where D = M3 +- 1, M [?] Z*},
     journal = {Acta Arithmetica},
     volume = {89},
     year = {1999},
     pages = {259-277},
     zbl = {0932.11010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav89i3p259bwm}
}

Christian Ballot. Strong arithmetic properties of the integral solutions of X³ + DY³ + D²Z³ - 3DXYZ = 1, where D = M³ ± 1, M ∈ ℤ*. Acta Arithmetica, Tome 89 (1999) pp. 259-277. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav89i3p259bwm/

[000] [Ba1] C. Ballot, Density of prime divisors of linear recurrences, Mem. Amer. Math. Soc. 551 (1995).

[001] [Ba2] C. Ballot, Group structure and maximal division for cubic recursions with a double root, Pacific J. Math. 173 (1996), 337-355. | Zbl 0866.11007

[002] [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. | Zbl 0979.11007

[003] [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. | Zbl 0139.27501

[004] 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. | Zbl 0569.10003

[005] J. C. Lagarias, Sets of primes determined by systems of polynomial congruences, Illinois J. Math. 27 (1983), 224-237. | Zbl 0507.12004

[006] [Lax] R. R. Laxton, On groups of linear recurrences I, Duke Math. J. 26 (1969), 721-736. | Zbl 0226.10010

[007] [Le1] D. H. Lehmer, On the multiple solutions of the Pell equation, Ann. of Math. (2) 30 (1929), 66-72.

[008] [Le2] D. H. Lehmer, On Lucas's test for the primality of Mersenne's numbers, J. London Math. Soc. 10 (1935), 162-165. | Zbl 0012.10301

[009] [Lu] E. Lucas, Théorie des fonctions numériques simplement périodiques, Amer. J. Math. 1 (1878), 184-240, 289-321.

[010] [Ma] G. B. Mathews, On the arithmetical theory of the form x³ + ny³ + n²z³ - 3nxyz, Proc. London Math. Soc. 21 (1890), 280-287. | Zbl 22.0220.01

[011] [Mo] P. Moree, On the prime density of Lucas sequences, J. Théor. Nombres Bordeaux 8 (1996), 449-459. | Zbl 0873.11058

[012] [M-S] P. Moree and P. Stevenhagen, Prime divisors of Lucas sequences, Acta Arith. 82 (1997), 403-410. | Zbl 0913.11048

[013] [Na] T. Nagell, Über die Einheiten in reinen kubischen Zahlkörpern, Skr. Norske Vid. Akad. Oslo Mat.-Naturv. Klasse 11 (1923), 1-34. | Zbl 49.0111.03

[014] [Wa1] M. Ward, The maximal prime divisors of linear recurrences, Canad. J. Math. 6 (1954), 455-462. | Zbl 0056.04106

[015] [Wa2] M. Ward, The prime divisors of Fibonacci numbers, Pacific J. Math. 11 (1961), 379-386. | Zbl 0112.26904

[016] [We] A. E. Western, On Lucas's and Pepin's tests for primeness of Mersenne numbers, J. London Math. Soc. 7 (1932), 130-137. | Zbl 0004.24402

[017] [Wi1] H. C. Williams, A generalization of the Lucas functions, unpublished Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1969.

[018] [Wi2] H. C. Williams, On a generalization of the Lucas functions, Acta Arith. 20 (1972), 33-51. | Zbl 0248.10011

[019] [Wi3] H. C. Williams, Edouard Lucas and Primality Testing, Canad. Math. Soc. Ser. Monographs Adv. Texts, Wiley, 1998.