On sums of two rational cubes (Q851292)

The authors study the Diophantine equation \[ x^3 + y^3 = az^3, \] for \(a = pq\), \(p\), \(q\) prime numbers satisfying \(p\equiv 2 \bmod 3\) and \(q\equiv 1\bmod 3\), such that \(4q-p^2 \not\equiv 3 \bmod 9\) and \(p\) is a non-cubic residue modulo \(q\). Using descent arguments it is shown that the equation has no nonzero solution. This article seems to be identical to an earlier one by the first author [Mosc. Univ. Math. Bull. 56, No. 1, 20--23 (2001); translation from Vestn. Mosk. Univ., Ser. I 2001, No. 1, 19--22 (2001; Zbl 1052.11021)].