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An exponential diophantine equation

Published online by Cambridge University Press:  17 April 2009

Maohua Le
Maohua Le
Affiliation:
Department of Mathmatics, Zhanjiang Normal College, Postal code 524048, Zhanjiang, Guangdong, People's Republic of China
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Abstract

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Let p be an odd prime with p > 3. In this paper we give all positive integer solutions (x, y, m, n) of the equation x2 + p2m = yn, gcd (x, y) = 1, n > 2 satisfying 2 | n of 2 ∤ n and p ≢ (−1)(p−1)/2(mod 4n.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 1 , August 2001 , pp. 99 - 105
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Arif, S.A. and Muriefah, F.S.A., ‘The diophantine equation x 2 + 3m = yn’, Internat. J. Math. Math. Sci. 21 (1998), 619–620.CrossRefGoogle Scholar
[2]Bilu, Y., Hanrot, G. and Voutier, P.M., ‘Existence of primitive divisors of Lucas and Lehmer numbers’, J. Reine Angew. Math. (to appear).Google Scholar
[3]Cohn, J.H.E., ‘The diophantine equation x 2 + 2k = yn’, Arch. Math. (Basel) 59 (1992), 341–344.CrossRefGoogle Scholar
[4]Le, M.-H., ‘On the diophantine equation (xm + 1)(xn + 1) = y 2’, Acta Arith. 82 (1997). 17–26.CrossRefGoogle Scholar
[5]Le, M.-H., ‘Diophantine equation x 2 + 2m = yn’, Chinese Sci. Bull. 2 (1997), 1515–1517.Google Scholar
[6]Le, M.-H., ‘On Cohn's conjecture concerning the diophantine equation x 2 + 2m = yn’, Arch. Math. (Basel) (to appear).Google Scholar
[7]Ljunggren, W., ‘Zur Theorie der Gleichung x 2 + 1 = Dy 4’, Avh. Norske Vid. Akad. Oslo 5 (1942), 1–27.Google Scholar
[8]Ljunggren, W., ‘Ein satz über die Diophantische gleichung Ax 2 − By 4 = C (C = 1, 2, 4)’, Tolfte Skandinaviska Matematikerkongressen, Lund (1954), 188–194.Google Scholar
[9]Luca, F., ‘On a diophantine equation’, Bull. Austral. Math. Soc. 61 (2000), 241–246.CrossRefGoogle Scholar
[10]Lucas, E., ‘Théorie des functions numériques simplement périodiques’, Amer. J. Math. 1 (1878), 184–240, 289–321.CrossRefGoogle Scholar
[11]Mordell, L.J., Diophantine equations (Academic Press, London, 1969).Google Scholar
[12]Nagell, T., ‘Contributions to the theory of a category of diophantine equations of the second degree with two unknowns’, Nova Acta Soc. Sci. Upsal. (4) 16 (1954), 1–38.Google Scholar
[13]Störmer, C., ‘L'équation m arc tan (1/x) + n arc tan (1/y) = kπ/4’, Bull. Soc. Math. France 27 (1899), 160–170.CrossRefGoogle Scholar
[14]Voutier, P.M., ‘Primitive divisors of Lucas and Lehmer sequences’, Math. Comp. 64 (1995), 869–888.Google Scholar