On the diophantine equation \(x^3+y^3+z^3=nxyz\) (Q1280359)

For the equation (1) \(x^3+y^3+z^3=nxyz\) the author proves the following assertions. (a) If \(n\in \{4k,8k-1,2^{2m+1}(2k-1)+3\}\) for \(m\) and \(k\) positive integers, then the equation (1) does not have positive integer solutions. (b) For any \(i\in \{1,2,3,5,6\}\) there exist infinitely many positive integers \(n\) of the form \(8k+i \) for which the equation \(x^3+y^3+1=nxy\) has positive integer solutions. (c) The equation \(x^3+64y^3+1=8xyz\) does not have positive integer solutions.