Fermat's `primitive solutions' and some arithmetic of elliptic curves
Already Fermat studied elliptic curves \(E: y^ 2 = ax^ 4 + bx^ 3 + cx^ 2 + dx + e\) and discovered a method to find rational points on such curves: starting from ``primitive'' points, Fermat described how to construct infinitely many ``derived points''. These derived points turned out to belong to the subgroup generated by the primitive points of the Mordell-Weil group of \(E(\mathbb{Q})\). In particular, Fermat discovered six primitive points of \(E_ 1: y^ 2 = x^ 4 + 4x^ 3 + 10x^ 2 +20x + 1\) and Nagell posed in 1929 the question whether these points are independent in the Mordell-Weil group of \(E_ 1(\mathbb{Q})\). By using the presently known theory for elliptic curves and available software, the author shows (unconditionally) that the Mordell-Weil group of \(E_ 1(\mathbb{Q})\) has rank 2 and that it is generated by two of the primitive points discovered by Fermat.
scientific article; zbMATH DE number 469233
scientific article