On the Diophantine equation A x 2 + 2 2 m = y n

Fadwa S. Abu Muriefah

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 25, 1-9

Abstract:

Let h denote the class number of the quadratic field ℚ ( − A ) for a square free odd integer A > 1 , and suppose that n > 2 is an odd integer with ( n , h ) = 1 and m > 1 . In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with x and y odd. The case n = 3 is solved completely. A result of E. Brown for A = 3 is improved and generalized to the case where A is a prime ≢ 7 ( mod 8 ) .

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:312951

DOI: 10.1155/S0161171201004835

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