The diophantine equation x² + C = yⁿ
The diophantine equation x² + 19 = yⁿ
The Diophantine equation x⁴ - Dy² = 1, II
The diophantine equation , II
The Diophantine equation xⁿ = Dy² + 1
The Diophantine equation
It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.
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