On some generalized Lebesgue-Nagell equations. (English) Zbl 1219.11059

Zhu, Huilin; Le, Maohua

J. Number Theory 131, No. 3, 458-469 (2011).

This paper continues the study of the Lebesgue-Nagell equation \(x^2+D=y^n\), where \(n\geq 3\). Here the Authors consider the special case \(D=q^m\) where \(q\) is a prime number \({}\geq 11\) and they solve completely this exponential Diophantine equation when the ring of integers of the quadratic field \({\mathbb Q}(\sqrt{-q})\) is principal, that is when \(q\in\{11,19,43,67,163\}\).
The proof combines deep (obtained by modular method and Baker’s theory) and elementary arguments.

Reviewer: Maurice Mignotte (Strasbourg)

MSC:

11D61

Exponential Diophantine equations

Keywords:

generalized Lebesgue-Nagell equation; ternary Diophantine equation; primitive divisor

References:

[1]	Abu Muriefah, F.S., On the Diophantine equation \(x^2 + 5^{2 k} = y^n\), Demonstratio math., 39, 2, 285-289, (2006) · Zbl 1100.11013
[2]	Abu Muriefah, F.S.; Arif, S.A., The Diophantine equation \(x^2 + 5^{2 k + 1} = y^n\), Indian J. pure appl. math., 30, 3, 229-231, (1999) · Zbl 0940.11017
[3]	Arif, S.A.; Abu Muriefah, F.S., On the Diophantine equation \(x^2 + 2^k = y^n\), Int. J. math. math. sci., 20, 2, 299-304, (1997) · Zbl 0881.11038
[4]	Arif, S.A.; Abu Muriefah, F.S., On a Diophantine equation, Bull. austral. math. soc., 57, 189-198, (1998) · Zbl 0905.11018
[5]	Arif, S.A.; Abu Muriefah, F.S., The Diophantine equation \(x^2 + 3^m = y^n\), Int. J. math. math. sci., 21, 610-620, (1998) · Zbl 0905.11017
[6]	Arif, S.A.; Muriefah, F.S.A., On the Diophantine equation \(x^2 + q^{2 k + 1} = y^n\), J. number theory, 95, 95-100, (2002) · Zbl 1037.11021
[7]	Bennett, M.A.; Skinner, C.M., Ternary Diophantine equation via Galois representations and modular forms, Canad. J. math., 56, 23-54, (2004) · Zbl 1053.11025
[8]	Bérczes, A.; Pink, I., On the Diophantine equation \(x^2 + p^{2 k} = y^n\), Arch. math., 91, 505-517, (2008) · Zbl 1175.11018
[9]	Bilu, Y.; Hanrot, G.; Voutier, F.M., Existence of primitive divisors of Lucas and Lehmer numbers, J. reine angew. math., 539, 75-122, (2001), With an appendix by M. Mignotte · Zbl 0995.11010
[10]	Bugeaud, Y., On some exponential Diophantine equations, Monatsh. math., 132, 93-97, (2001) · Zbl 1014.11023
[11]	Bugeaud, Y.; Mignotte, M.; Siksek, S., Classical and modular approached to exponential Diophantine equations II. the Lebesgue-nagell equation, Compos. math., 142, 31-62, (2006) · Zbl 1128.11013
[12]	Cohn, J.H.E., The Diophantine equation \(x^2 + 2^k = y^n\), Arch. math. (basel), 59, 4, 341-344, (1992) · Zbl 0770.11019
[13]	Cohn, J.H.E., The Diophantine equation \(x^2 + 2^k = y^n\), II, Int. J. math. math. sci., 22, 3, 459-462, (1999) · Zbl 0960.11025
[14]	Gebel, J.; Pethö, A.; Zimmer, H.G., On Mordell’s equation, Compos. math., 110, 335-367, (1998) · Zbl 0899.11013
[15]	Lal, M.; Jones, M.F.; Blundon, W.J., Numerical solutions of \(y^3 - x^2 = k\), Math. comp., 20, 322-325, (1996) · Zbl 0136.32703
[16]	Le, M., On Cohn’s conjecture concerning the Diophantine equation \(x^2 + 2^m = y^n\), Arch. math. (basel), 78, 1, 26-35, (2002) · Zbl 1006.11013
[17]	Lebesgue, V.A., Sur l’impossibilite én nombres entiers de l’équation \(x^m = y^2 + 1\), Nouv. ann. math., 9, 1, 178-181, (1850)
[18]	London, J.; Finkelstein, M., On Mordell’s equation \(y^2 - k = x^3\), (1973), Bowling Green State University Bowling Green, OH · Zbl 0276.10009
[19]	Luca, F., On a Diophantine equation, Bull. austral. math. soc., 61, 241-246, (2000) · Zbl 0997.11027
[20]	Luca, F., On the Diophantine equation \(x^2 + 7^{2 k} = y^n\), Fibonacci quart., 45, 4, 322-326, (2007) · Zbl 1221.11091
[21]	Saradha, N.; Srinivasan, A., Solutions of some generalized Ramanujan-nagell equations, Indag. math., 17, 1, 103-114, (2006) · Zbl 1110.11012
[22]	Siksek, S.; Cremona, J.E., On the Diophantine equation \(x^2 + 7 = y^m\), Acta arith., 109, 2, 143-149, (2003) · Zbl 1026.11043
[23]	Tao, L., On the Diophantine equation \(x^2 + 3^m = y^n\), Electr. J. comm. number theory, 8, 1-7, (2008) · Zbl 1210.11048
[24]	Tao, L., On the Diophantine equation \(x^2 + 5^m = y^n\), Ramanujan J., 19, 325-338, (2009) · Zbl 1191.11008