A note on the number of solutions of the generalized Ramanujan-Nagell equation $x²-D = k^n$

Maohua Le

A note on the number of solutions of the generalized Ramanujan-Nagell equation

x ² - D = k^{n}

Maohua Le

Acta Arithmetica, Tome 76 (1996), p. 11-18 / Harvested from The Polish Digital Mathematics Library

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Publié le : 1996-01-01

Zbl 0869.11028

EUDML-ID : urn:eudml:doc:206929

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     author = {Maohua Le},
     title = {A note on the number of solutions of the generalized Ramanujan-Nagell equation $x$^2$-D = k^n$
            },
     journal = {Acta Arithmetica},
     volume = {76},
     year = {1996},
     pages = {11-18},
     zbl = {0869.11028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav78i1p11bwm}
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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, Tome 76 (1996) pp. 11-18. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav78i1p11bwm/

Bibliographie

[000] [1] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1263-1264. | Zbl 0093.04703

[001] [2] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1451-1452. | Zbl 0093.04703

[002] [3] F. Beukers, On the generalized Ramanujan-Nagell equation I, Acta Arith. 38 (1981), 389-410. | Zbl 0371.10014

[003] [4] F. Beukers, On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 113-123. | Zbl 0377.10012

[004] [5] E. Brown, The diophantine equation of the form $x ² + D = y^{n}$ , J. Reine Angew. Math. 274/275 (1975), 385-389. | Zbl 0303.10014

[005] [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.

[006] [7] L.-K. Hua, Introduction to Number Theory, Springer, Berlin, 1982.

[007] [8] M.-H. Le, On the generalized Ramanujan-Nagell equation $x ² - D = p^{n}$ , Acta Arith. 58 (1991), 289-298.

[008] [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.

[009] [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.

[010] [11] M.-H. Le, Some exponential diophantine equations I: The equation $D ₁ x ² - D ₂ y ² = λ k^{z}$ , J. Number Theory 55 (1995), 209-221.

[011] [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.

[012] [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.

[013] [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. | Zbl 0586.10011

[014] [15] T.-J. Xu and M.-H. Le, On the diophantine equation $D ₁ x ² + D ₂ = k^{n}$ , Publ. Math. Debrecen 47 (1995), 293-297.