×

On the equation \(Y^2 = X^5 + k\). (English) Zbl 1210.11045

Summary: We show that there are infinitely many nonisomorphic curves \(Y^2 = X^5 + k\), \(k\in\mathbb Z\), possessing at least twelve finite points \(k>0\), and at least six finite points for \(k<\). We also determine all rational points on the curve \(Y^2=X^5-7\).

MSC:

11D41 Higher degree equations; Fermat’s equation
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
PDF BibTeX XML Cite
Full Text: DOI Euclid