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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 titlehas no solution in positive integers x and y if n has anyprime 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)

international journal of mathematics and mathematical sciences

On the Diophantine equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="$Ax^2 + 2^{2m} = y^n" id="E1"><mml:mi>A</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mn>2</mml:mn><mml:mrow><mml:mn>2</mml:mn><mml:mi>m</mml:mi></mml:mrow></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:math>

On the Diophantine equationAx2+22m=yn