## 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

### Keywords:

fifth powers; genus two curve; elliptic curve
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