On the diophantine equation \(2x^ 3+1=py^ 2\)
Let \(p\equiv 1 (mod 4)\) be a prime number such that the field \(K={\mathbb{Q}}(\sqrt{-3p})\) has class number 2. Then the diophantine (hyperelliptic) equation (1) \(X^ 6+4=pY^ 2\) has an integral solution [\textit{C. Meyer}, Symp. Math. 15, 365-387 (1975; Zbl 0329.12007), and the reviewer, J. Reine Angew. Math. 339, 27-81 (1983; Zbl 0495.12005)]. If \(p\equiv 5 (mod 8)\) then the equation (1) can be easily solved. In case \(p\equiv 1 (mod 8)\) the field \(K={\mathbb{Q}}(\sqrt{-3p})\) has class number 2 only for \(p=17,41,89\) and solutions of (1) are \((p,| X|,| Y|)=(17,2,2), (41,4,10), (89,10,106).\) Using the p-adic method it is proved in the paper under review that for \(p=17,41,89\) no other solutions exist.
scientific article; zbMATH DE number 3943913