On the Diophantine equation Ax2+22m=yn
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 congru...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201004835 |
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| Summary: | 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(mod8)
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| ISSN: | 0161-1712 1687-0425 |