The Diophantine equation \(x^3+y^3=pqz^3\) (Q1851499)

The main result of the paper is as follows. Theorem: Let \(p\) and \(q\) be prime numbers, \(p\equiv 2\pmod 3\), \(q\equiv 1\pmod 3\), \(4q-p^2\not\equiv 3\pmod 9\). If \(p\) is not a cubic residue modulo \(q\), then the equation \(x^3+y^3= pqz^3\) has no solutions in nonzero integers \(x\), \(y\), \(z\).