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

• Autorzy: Maohua Le
• Języki publikacji: EN

Kategorie tematyczne

• 11D61: Exponential equations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1996-1997
• Tom: 78
• Numer: 1
• Strony: 11-18

Daty

wydano

1996

otrzymano

1995-08-29

poprawiono

1996-03-18

Twórcy

autor

Maohua Le

• Department of Mathematics, Zhanjiang Teachers College, 524048 Zhanjiang, Guangdong, P.R. China

Bibliografia

• [1] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1263-1264.
• [2] R. Apéry, Sur une équation diophantienne, C. R. Acad. Sci. Paris Sér. A 251 (1960), 1451-1452.
• [3] F. Beukers, On the generalized Ramanujan-Nagell equation I, Acta Arith. 38 (1981), 389-410.
• [4] F. Beukers, On the generalized Ramanujan-Nagell equation II, Acta Arith. 39 (1981), 113-123.
• [5] E. Brown, The diophantine equation of the form $x²+D=y^n$, J. Reine Angew. Math. 274/275 (1975), 385-389.
• [6] X.-G. Chen and M.-H. Le, On the number of solutions of the generalized Ramanujan-Nagell equation $x²-D=k^n$, Publ. Math. Debrecen, to appear.
• [7] L.-K. Hua, Introduction to Number Theory, Springer, Berlin, 1982.
• [8] M.-H. Le, On the generalized Ramanujan-Nagell equation $x²-D=p^n$, Acta Arith. 58 (1991), 289-298.
• [9] M.-H. Le, On the number of solutions of the generalized Ramanujan-Nagell equation $x²-D=2^{n+2}$, Acta Arith. 60 (1991), 149-167.
• [10] M.-H. Le, Sur le nombre de solutions de l'équation diophantienne $x²+D=p^n$, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 135-138.
• [11] M.-H. Le, Some exponential diophantine equations I: The equation $D₁x²-D₂y²=λk^z$, J. Number Theory 55 (1995), 209-221.
• [12] V. A. Lebesgue, Sur l'impossibilité, en nombres entiers, de l'équation $x^m=y²+1$, Nouv. Ann. Math. (1) 9 (1850), 178-181.
• [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] N. Tzanakis and J. Wolfskill, On the diophantine equation $y²=4q^n+4q+1$, J. Number Theory 23 (1986), 219-237.
• [15] T.-J. Xu and M.-H. Le, On the diophantine equation $D₁x²+D₂=k^n$, Publ. Math. Debrecen 47 (1995), 293-297.