The Diophantine equation \(x^3 + 3y^3 = 2^n\) (Q1168351)

Consider the equation in the title with \(xy\) odd. By studying this equation in the field \(\mathbb Q(\vartheta)\), where \(\vartheta = \sqrt [3]{3}\) (real), the author first proves that there are no solutions if \(n\) is even, \(n >2\). Then he extends his considerations to odd values of \(n(>2)\) and, working in \(\mathbb Q(\vartheta)\) modulo powers of 61, derives necessary conditions for \((x,y,n)\) to be a solution. In particular, there are only a finite number of solutions and, for a fixed \(x\) [or \(y\)], at most one solution \((y,n)\) [resp. \((x,n)\)].