A note on the number of solutions of the generalized Ramanujan-Nagell equation x ² - D = k n

Maohua Le

Acta Arithmetica (1996)

  • Volume: 78, Issue: 1, page 11-18
  • ISSN: 0065-1036

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Maohua Le. "A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$." Acta Arithmetica 78.1 (1996): 11-18. <http://eudml.org/doc/206929>.

@article{MaohuaLe1996,
author = {Maohua Le},
journal = {Acta Arithmetica},
keywords = {exponential diophantine equations; number of solutions of the generalized Ramanujan-Nagell equation},
language = {eng},
number = {1},
pages = {11-18},
title = {A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$},
url = {http://eudml.org/doc/206929},
volume = {78},
year = {1996},
}

TY - JOUR
AU - Maohua Le
TI - A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$
JO - Acta Arithmetica
PY - 1996
VL - 78
IS - 1
SP - 11
EP - 18
LA - eng
KW - exponential diophantine equations; number of solutions of the generalized Ramanujan-Nagell equation
UR - http://eudml.org/doc/206929
ER -

References

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  1. [1] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1263-1264. Zbl0093.04703
  2. [2] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1451-1452. Zbl0093.04703
  3. [3] F. Beukers, On the generalized Ramanujan-Nagell equation I, Acta Arith. 38 (1981), 389-410. Zbl0371.10014
  4. [4] F. Beukers, On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 113-123. Zbl0377.10012
  5. [5] E. Brown, The diophantine equation of the form x ² + D = y n , J. Reine Angew. Math. 274/275 (1975), 385-389. Zbl0303.10014
  6. [6] X.-G. Chen and M.-H. Le, On the number of solutions of the generalized Ramanujan-Nagell equation x ² - D = k n , Publ. Math. Debrecen, to appear. 
  7. [7] L.-K. Hua, Introduction to Number Theory, Springer, Berlin, 1982. 
  8. [8] M.-H. Le, On the generalized Ramanujan-Nagell equation x ² - D = p n , Acta Arith. 58 (1991), 289-298. 
  9. [9] M.-H. Le, On the number of solutions of the generalized Ramanujan-Nagell equation x ² - D = 2 n + 2 , Acta Arith. 60 (1991), 149-167. 
  10. [10] M.-H. Le, Sur le nombre de solutions de l’équation diophantienne x ² + D = p n , C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 135-138. 
  11. [11] M.-H. Le, Some exponential diophantine equations I: The equation D x ² - D y ² = λ k z , J. Number Theory 55 (1995), 209-221. 
  12. [12] V. A. Lebesgue, Sur l’impossibilité, en nombres entiers, de l’équation x m = y ² + 1 , Nouv. Ann. Math. (1) 9 (1850), 178-181. 
  13. [13] T. Nagell, Contributions to the theory of a category of diophantine equations of the second degree with two unknowns, Nova Acta R. Soc. Sc. Uppsal. (4) 16 (1954), No. 2. 
  14. [14] N. Tzanakis and J. Wolfskill, On the diophantine equation y ² = 4 q n + 4 q + 1 , J. Number Theory 23 (1986), 219-237. Zbl0586.10011
  15. [15] T.-J. Xu and M.-H. Le, On the diophantine equation D x ² + D = k n , Publ. Math. Debrecen 47 (1995), 293-297. 

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