Letter to the editor: ''Diophantine equation \(x^ 3-y^ 2=k\) and Hall's conjecture'' (Q1059656)

In the article mentioned in the title [Mat. Zametki 32, 273-275 (1982; Zbl 0499.10016)], there is an inaccuracy, whose removal improves by a factor of 2 the constant in the right-hand side of (5). It follows from the text of the article that on p. 274 we should have in the right-hand side of Eq. (9) -216 \(P_ m+432\), and hence we have a stronger form of the theorem. After the publication of this article, the author noticed that the constant in the theorem could again be significantly decreased, in order to give the following theorem. The inequalities \[ 0<| x^ 3-y^ 2| <0.97 | x|^{0,5} \] have infinitely many solutions in integers x and y.