On the diophantine equation $x^2 - p^m = ±y^n$

Yann Bugeaud

ArticleOriginal scientific text

Title

On the diophantine equation $x^{2} - p^{m} = \pm y^{n}$

Authors ¹

Affiliations

U.F.R. de mathématiques, Université Louis Pasteur, 7, rue René Descartes, 67084 Strasbourg, France

Y. Bugeaud, On the diophantine equation $x^{2} - 2^{m} = \pm y^{n}$ , Proc. Amer. Math. Soc., to appear.
Y. Bugeaud et M. Laurent, Minoration effective de la distance p-adique entre puissances de nombres algébriques, J. Number Theory 61 (1996), 311-342
A. Faisant, L'équation diophantienne du second degré, Hermann, Paris, 1991.
Y. D. Guo and M. H. Le, A note on the exponential diophantine equation $x^{2} - 2^{m} = y^{n}$ , Proc. Amer. Math. Soc. 123 (1995), 3627-3629.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1990.
C. Ko, On the diophantine equation $x^{2} = y^{n} + 1$ , xy ≠ 0, Sci. Sinica 14 (1965), 457-460.
S. V. Kotov, Über die maximale Norm der Idealteiler des Polynoms $a x^{m} + b y^{n}$ mit den algebraischen Koeffizienten, Acta Arith. 31 (1976), 219-230.
M. Laurent, M. Mignotte et Y. Nesterenko, Formes linéaires en deux logarithmes et déterminants d'interpolation, J. Number Theory 55 (1995), 285-321.
M. H. Le, On the generalized Ramanujan-Nagell equation, III, Dongbei Shuxue 4 (1988), 180-184.
M. H. Le, The diophantine equation $x^{2} + D^{m} = p^{n}$ , Acta Arith. 52 (1989), 255-265.
M. H. Le, Applications of Baker's method, IV, J. Changsha Railway Inst. 9 (1991), no. 2, 87-92.
M. H. Le, A note on the diophantine equation $\frac{x^{m} - 1}{x - 1} = y^{n}$ , Acta Arith. 64 (1993), 19-28.
M. H. Le, Upper bounds for class number of real quadratic fields, ibid. 68 (1994), 141-144.
M. H. Le, Some exponential diophantine equations, I, J. Number Theory 55 (1995), 209-221.
T. N. Shorey, A. J. Van der Poorten, R. Tijdeman and A. Schinzel, Applications of the Gel'fond-Baker method to diophantine equations, in: Transcendence Theory: Advances and Applications, Academic Press, London, 1977, 59-78.
M. Toyoizumi, On the diophantine equation $x^{2} + D^{m} = p^{n}$ , Acta Arith. 42 (1983), 303-309.

Title

On the diophantine equation x2-pm=±yn

Affiliations

Bibliography

On the diophantine equation $x^{2} - p^{m} = \pm y^{n}$