Examples of Mordell’s Equation
نویسنده
چکیده
The equation y2 = x3 +k, for k ∈ Z, is called Mordell’s equation on account of Mordell’s long interest in it throughout his life. A natural number-theoretic task is the description of all rational and integral solutions to such an equation, either qualitatively (decide if there are finitely or infinitely many solutions in Z or Q) or quantitatively (list or otherwise conveniently describe all such solutions). Mordell, in 1922, proved that for each k ∈ Z, the equation y2 = x3 + k has only finitely many integral solutions. The rational solutions present a different story: there may be finitely many or infinitely many, depending on the integer k. Whether or not there are infinitely many rational solutions is connected to one of the most outstanding open problems in number theory, the Birch and Swinnerton–Dyer conjecture. Here we will describe all the integral solutions to Mordell’s equation for some selected values of k, and make a few comments at the end about rational solutions.
منابع مشابه
18.782 Arithmetic Geometry Lecture Note 25
In the last lecture we proved that the torsion subgroup of the rational points on an elliptic curve E/Q is finite. In this lecture we will prove a special case of Mordell’s theorem, which states that E(Q) is finitely generated. By the structure theorem for finitely generated abelian groups, this implies E(Q) ' Z ⊕ T, where Zr is a free abelian group of rank r, and T is the (necessarily finite) ...
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تاریخ انتشار 2009