# zbMATH — the first resource for mathematics

On the number of the integral points of elliptic curves. (Russian. English summary) Zbl 0879.11028
Let $$\mathcal F: y^2=x^3+rx+s$$ be an elliptic curve such that $$r,s\in \mathbb{Z}$$, $$4r^3+27s^2\neq 0$$. Suppose that the congruences $$r\equiv 0\bmod c^4$$, $$s\equiv 0\bmod c^6$$ are not satisfied for any $$c\in\mathbb{Z}$$, $$c>1$$, and that $$\mathcal F(\mathbb{Q})/\mathcal F_{\text{tors}}(\mathbb{Q})\cong\mathbb{Z}$$. The author proves that if $$\mathcal F_{\text{tors}}(\mathbb{Q})$$ is trivial, then there are at most 12 integral points on $$\mathcal F$$. Using the curve $$y^2=x^3-x+1$$ as an example, he shows that the result is best possible. He further states that, if $$\mathcal F_{\text{tors}}(\mathbb{Q})$$ is nontrivial, then the number of integral points is at most 18, but no outline of a proof is given.
##### MSC:
 11G05 Elliptic curves over global fields 11D25 Cubic and quartic Diophantine equations 14H52 Elliptic curves
##### Keywords:
elliptic curve; number of integral points