On Bachet's diophantine equation \(x^ 3=y^ 2+k\)
Using Eisenstein's law of cubic reciprocity we investigate cases in which \(x^ 3=y^ 2+k\) is unsolvable in the ring of rational integers \({\mathbb{Z}}\). In particular we show, that for all primes \(p\not\equiv \pm 1\quad(mod 9),\) \(p\neq 3\), the equation \(x^ 3=y^ 2+3p\quad(p\pm 9)\) has no solutions in \({\mathbb{Z}}\).
unsolvability in the ring of rational integers
scientific article; zbMATH DE number 3867421