The complete solution in integers of \(x^ 3+3y^ 3=2^ n\)
The author has solved the diophantine equation of the title for even n in [ibid. 15, 376-387 (1982; Zbl 0493.10021)]. Concerning some arithmetics of the fields \({\mathbb{Q}}(\delta)\) with \(\delta^ 3=3\) and 6 (\(\delta\) real) he gets the complete solution with x y odd: \((x,y,n)=(- 1,1,1),\quad(1,1,2),\quad(-7,5,5),\quad(5,1,7).\) The article (communicated by D. J. Lewis) ends with the following problem (proposed by D. J. Lewis): Let N(p) denote the number of integer solutions of \(x^ 3+3y^ 3=p^ n\) with \((x,y)=1\). Decide if N(p) is bounded as \(p\to \infty.\) \(\{\) Reviewer's comment: N(p) is bounded, and explicit estimations are given by \textit{D. J. Lewis} and \textit{K. Mahler} [Acta Arith. 6, 333-363 (1961; Zbl 0102.036)]. For better estimations, cf. \textit{J. H. Evertse} [Upper bounds for the numbers of solutions of diophantine equations (1983; Zbl 0517.10016), p. 74].\(\}\)
scientific article; zbMATH DE number 3863286