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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.
11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
14H52 Elliptic curves