turkmath.org

A well-known theorem of Siegel states that any elliptic curve $E/\mathbb{Q}$ has only finitely many integral points. Lang conjectured that the number of integral points on a quasi-minimal model of an elliptic curve should be bounded solely in terms of the rank of the group of rational points. Silverman proved Lang's conjecture for the curves with at most a fixed number of primes dividing the denominator of the $j$-invariant. Using more explicit methods, Silverman and Gross compute the dependence of the bounds on the various constants. In the case of curves of rank 1, techniques of Ingram on multiples of integral points enable one to prove much better bounds for special families of elliptic curves. In this talk, we investigate the integral points on Mordell curves of rank 1.