Integer points on elliptic curves (Q5891878)

Let \(f(x)\) be a polynomial of degree \(d\) with integer coefficients and \(c\) a non-zero integer. Assume that \(c y^2 - f(x)\) is absolutely irreducible. By \textit{C. L. Siegel}'s theorem [Abh. Preuß. Akad. Wiss., Phys.-Math. Kl. 1929, No. 1, 70 S. (1929; JFM 56.0180.05)], we know that there are only finitely many integral points on the curve \(c y^2 = f(x)\). Siegel's method does not provide any local bound. The author uses \textit{P. X. Gallagher}'s sieve [Mathematika 14, 14--20 (1967; Zbl 0163.04401)] to obtain upper bounds for the number of integral points of bounded size on the curve \(c y^2 = f(x)\). For real numbers \(X_{0}\) and \(X \geq 1\), define \(N_{f}(X ; X_{0})\) to be the number of integral points \((x , y)\) with \(X_{0} < x \leq X_{0} + X\) and \(c y^2 = f(x)\). The author shows that NEWLINE\[NEWLINE N_{f}(X; X_{0})\ll X^{1/2} NEWLINE\]NEWLINE where the implicit constant depends at most on \(d\), the degree of \(f(x)\). In particular, the author concludes that the number of positive integer points on the Mordell equation NEWLINE\[NEWLINE x^3 + y^2 =n, NEWLINE\]NEWLINE where \(n \in \mathbb{N}\), is \(\ll n^{1/6}\).