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

Abstract

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

18 pages