Publicado

2006-01-01

The Diophantine equation x^2 + 2 = y^n : a brief overview

Palabras clave:

Diophantine equations, . Baker’s method, 2000 Mathematics Subject Classification, Primary: 11D61 (en)

Descargas

Autores/as

  • Fadwa S. Abu Muriefah Girls College Of Education, Saudi Arabia
  • Yann Bugeaud Université Louis Pasteur, France

Abstract. We give a survey on recent results on the Diophantine equation x2 + c = yn.

Referencias

F. S. Abu Muriefah, On the Diophantine equation x^2 + 5^2k = y^n , Demo. Math. (To appear).

F. S. Abu Muriefah & S. A. Arif, On a Diophantine equation, Bull. Austral. Math. Soc. 5 7 (1998), 189-198.

F. S. Abu Muriefah & S. A. Arif, The Diophantine equation x^2 + 5^2k+1 = y^n, Indian J. Pure Appl. Math. 30 (1999), 229-231.

F. S. Abu Muriefah & S. A. Arif, The Diophantine equation x^2 + q^2k = y^n, Arab. J. Sci. Eng. Sect. A Sci. 26 (2001), 53-62.

S. A. Arif & F. S. Abu Muriefah, On the Diophantine equation x^2 + 2^k = y^n, Internat. J. Math. Math. Sci. 20 (1997), 299-304.

S. A. Arif & F. S. Abu Muriefah, The Diophantine equation x^2 + 3^m = y^n, Internat. J. Math. Math. Sci. 21 (1998), 619-S20.

S. A. Arif & F. S. Abu Muriefah, On the Diophantine equation x^2 + 2^k = y^n. II,, Arab. J. Math. Sci. 7 (2001), 67-71.

S. A. Arif & F. S. Abu Muriefah, On the Diophantine equation x^2 + q^2k+1 = y^n, J. Number Theory 95 (2002), 95-100.

M. A. Bennett & C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math. 56 [1] (2004), 23-54.

Yu. Bilu, On Le’s & Bugeaud’s papers about the equation a^x2 + b^2m-1 = 4c^p, Monatsh. Math. 1 37 (2002), 1-3

Yu. Bilu, G. Hanrot, & P. M. Voutier, with an appendix by M. Mignotte, Existence of primitive divisors of Lucas and Lehmer sequences, J. Reine Angew. Math. 539 (2001), 75-122.

Y. Bugeaud, M. Mignotte & S. Sik Sek, Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation, Compositio Math. 142 (2006), 31-62.

J. H. E. Cohn, The Diophantine equation x^2 + 2^k = y^n, Arch. Math. (Basel) 59 (1992), 341-344.

J. H. E. Cohn, The Diophantine equation x^2 + C = y^n, Acta Arith. 65 (1993), 367-381.

J. H. E. Cohn, The Diophantine equation x^2 + 2^k = y^n. II, Int. J. Math. Math. Sci. 22 (1999), 459-462.

J. H. E. C ohn, The Diophantine equation x^2 + C = y^n. II, Acta Arith. 109 (2003), 205-206.

Maohua Le, A note on the Diophantine equation x^2 + 7 = y^n, Glasgow Math. J. 39 (1997), 59-63.

M. Le, On Cohn’s conjecture concerning the Diophantine equation x^2 + 2^m = y^n, Arch. Math. (Basel) 78 [1] (2002), 26-35.

M. Le, On the Diophantine equation x^2 + p^2 = y^n, Publ. Math. Debrecen 63 (2003), 67-78.

V. A. Lebesgue, Sur l’impossibilité en nombres entiers de l’équation x^m = y^2 + 1, Nouvelles Annales des M athématiques 1 [9] (1850), 178-181.

J.-L. Lesage, Différence entre puissances et carrés d ’entiers, J. Number Theory 73 (1998), 390-425.

W. Ljunggren, Über einige Arcustangensgleichungen die auf intéressante unbestim mte Gleichungen führen, Ark. Mat. Astr. Fys. 29A [13] (1943).

F. Luca, On a Diophantine equation, Bull. Austral. Math. Soc. 61 (2000), 241-246.

F. Luca, On the equation x^2 + 2^a3^b= y^n, Int. J. Math. Math. Sci. 29 (2002), 239-244.

M. Mignotte, A kit on linear forms in three logarithms, IRMA, Strasbourg , to appear.

M. Mignotte & B. M. M. d e Weger, On the Diophantine equations x^2 + 74 = y^5 and x^2 + 86 = y^5, Glasgow Math. J. 38 (1996), 77-85.

T. Nagell, Sur l ’im possibilité de quelques équations à deux indéterminées, Norsk Mat. Forensings Skrifter 13 (1923), 65-82.

T. Nagell, Løsning til oppgave nr 2, 1943, s. 29, Nordisk Mat. Tidskr. 30 (1948), 62-64.

T. Nagell, Verallgemeinerung eines Fermatschen Satzes, Arch. Math. (Basel) 5 (1954), 153-159. [30] T. Nagell, Contributions to the theory of a category of Diophantine equations of the second degree with two unknowns, Nova Acta Regiae Soc. Sci. Upsaliensis 416 [2] (1955).

T. Nagell, Collected papers of Trygve Nagell. Vol. 1-4 Edited by Paulo Ribenboim. Queen’s Papers in Pure and Applied Mathematics, Queen’s University 121, Kingston, ON, 2002.

T. N. Shorey & R. Tijdeman, Exponential Diophantine equations, Cambridge University Press, Cambridge, 1986.

S. Siksek, On the Diophantine equation x^2 = y^p + 2^kz^p, J. Théor. Nombres Bordeaux 15 (2003), 839-846.

S. Siksek, The modular approach to Diophantine equations., In: Explicit Methods in Number Theory, Panoramas et Synthèses,, Société Mathématique De France, to appear.

S. Siksek & J. E. Cremona, On the Diophantine equation x^2 + 7 = y^m, Acta Arith. 109 (2003), 143-149.

V. G. Sprindžuk, Classical Diophantine equations, Lecture Notes in Mathematics 1559, Springer-Verlag, Berlin, 1993.

B. Sury, On the Diophantine equation x^2 + 2 = y^n , Arch. Math. (Basel) 74 (2000), 350-355.

Cómo citar

APA

Abu Muriefah, F. S. y Bugeaud, Y. (2006). The Diophantine equation x^2 + 2 = y^n : a brief overview. Revista Colombiana de Matemáticas, 40(1), 31–37. https://revistas.unal.edu.co/index.php/recolma/article/view/94655

ACM

[1]
Abu Muriefah, F.S. y Bugeaud, Y. 2006. The Diophantine equation x^2 + 2 = y^n : a brief overview. Revista Colombiana de Matemáticas. 40, 1 (ene. 2006), 31–37.

ACS

(1)
Abu Muriefah, F. S.; Bugeaud, Y. The Diophantine equation x^2 + 2 = y^n : a brief overview. rev.colomb.mat 2006, 40, 31-37.

ABNT

ABU MURIEFAH, F. S.; BUGEAUD, Y. The Diophantine equation x^2 + 2 = y^n : a brief overview. Revista Colombiana de Matemáticas, [S. l.], v. 40, n. 1, p. 31–37, 2006. Disponível em: https://revistas.unal.edu.co/index.php/recolma/article/view/94655. Acesso em: 2 feb. 2025.

Chicago

Abu Muriefah, Fadwa S., y Yann Bugeaud. 2006. «The Diophantine equation x^2 + 2 = y^n : a brief overview». Revista Colombiana De Matemáticas 40 (1):31-37. https://revistas.unal.edu.co/index.php/recolma/article/view/94655.

Harvard

Abu Muriefah, F. S. y Bugeaud, Y. (2006) «The Diophantine equation x^2 + 2 = y^n : a brief overview», Revista Colombiana de Matemáticas, 40(1), pp. 31–37. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94655 (Accedido: 2 febrero 2025).

IEEE

[1]
F. S. Abu Muriefah y Y. Bugeaud, «The Diophantine equation x^2 + 2 = y^n : a brief overview», rev.colomb.mat, vol. 40, n.º 1, pp. 31–37, ene. 2006.

MLA

Abu Muriefah, F. S., y Y. Bugeaud. «The Diophantine equation x^2 + 2 = y^n : a brief overview». Revista Colombiana de Matemáticas, vol. 40, n.º 1, enero de 2006, pp. 31-37, https://revistas.unal.edu.co/index.php/recolma/article/view/94655.

Turabian

Abu Muriefah, Fadwa S., y Yann Bugeaud. «The Diophantine equation x^2 + 2 = y^n : a brief overview». Revista Colombiana de Matemáticas 40, no. 1 (enero 1, 2006): 31–37. Accedido febrero 2, 2025. https://revistas.unal.edu.co/index.php/recolma/article/view/94655.

Vancouver

1.
Abu Muriefah FS, Bugeaud Y. The Diophantine equation x^2 + 2 = y^n : a brief overview. rev.colomb.mat [Internet]. 1 de enero de 2006 [citado 2 de febrero de 2025];40(1):31-7. Disponible en: https://revistas.unal.edu.co/index.php/recolma/article/view/94655

Descargar cita

Visitas a la página del resumen del artículo

23

Descargas

Los datos de descargas todavía no están disponibles.