On the Diophantine equation \(x^2+q^{2k+1}=y^n\). (English) Zbl 1037.11021

Arif, S. A.; Abu Muriefah, Fadwa S.

J. Number Theory 95, No. 1, 95-100 (2002).

It is proved that if \(q\) is an odd prime, \(q \not \equiv 7 \pmod 8\), \(n\) is an odd integer \(> 5\), \(n\) is not a multiple of 3 and \((h,n) = 1\), where \(h\) is the class number of the field \( \mathbb{Q}(\sqrt{-q})\), then the Diophantine equation \(x^2 + q^{2k+1} = y^n\) has exactly two families of solutions \((q,n,k,x,y)\).

Reviewer: Edward L. Cohen (Ottawa)

MSC:

11D41	Higher degree equations; Fermat’s equation
11D61	Exponential Diophantine equations

Keywords:

higher order Diophantine equations; Lucas sequence; primitive divisors; exponential Diophantine equation

Citations:

Zbl 0905.11017

References:

[1]	Arif, S.A.; Abu Muriefah, Fadwa S., The Diophantine equation x2+3m=yn, Internat. J. math. sci., 21, 619-620, (1998) · Zbl 0905.11017
[2]	Y. Bilu, G. Hanrot, and, P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, preprint. · Zbl 0995.11010
[3]	Cohn, J.H.E., The Diophantine equation x2+2k=yn, Arch. math. (basel), 59, 341-344, (1992) · Zbl 0770.11019
[4]	Cohn, J.H.E., The Diophantine equation x2+C=yn, Acta. arith., 65, 367-381, (1993) · Zbl 0795.11016
[5]	Luca, F., On a Diophantine equation, Bull. austral. math. soc., 61, 241-246, (2000) · Zbl 0997.11027
[6]	Nagell, T., Contributions to the theory of a category of Diophantine equations of second degree with two unknowns, Nova acta soc. sci., 16, 1-38, (1955)