On the Diophantine equation \(x^2+q^{2k+1}=y^n\). (English) Zbl 1037.11021

It is proved that if \(q\) is an odd prime, \(q \not \equiv 7 \pmod 8\), \(n\) is an odd integer \(> 5\), \(n\) is not a multiple of 3 and \((h,n) = 1\), where \(h\) is the class number of the field \( \mathbb{Q}(\sqrt{-q})\), then the Diophantine equation \(x^2 + q^{2k+1} = y^n\) has exactly two families of solutions \((q,n,k,x,y)\).


11D41 Higher degree equations; Fermat’s equation
11D61 Exponential Diophantine equations


Zbl 0905.11017
Full Text: DOI


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