On integer values of sum and product of three positive rational numbers

Abstract: In 1997 we proved that if $n$ is of the form $$ 4k, \quad 8k-1\quad {\rm or} \quad 2^{2m+1}(2k-1)+3, $$ where $k,m\in \mathbb N,$ then there are no positive rational numbers $x,y,z$ satisfying $$ xyz = 1, \quad x+y+z = n. $$ Recently, N. X. Tho proved the following statement: let $a\in\mathbb N$ be odd and let either $n\equiv 0\pmod 4$ or $n\equiv 7\pmod 8$. Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers $x,y,z.$ A representative example of our result is the following statement: assume that $a,n\in\mathbb N$ are such that at least one of the following conditions hold: $\bullet$ $n\equiv 0\pmod 4$ $\bullet$ $n\equiv 7\pmod 8 $ $\bullet$ $a\equiv 0\pmod 4$ $\bullet$ $a\equiv 0\pmod 2$ and $n\equiv 3\pmod 4$ $\bullet$ $a^2n^3=2^{2m+1}(2k-1)+27$ for some $k,m\in \mathbb N.$ Then the system of equations $$ xyz = a, \quad x+y+z = an. $$ has no solutions in positive rational numbers $x,y,z.$