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

Abstract

In 1997 we proved that if n is of the form $$\begin{aligned} 4k, \quad 8k-1\quad {\textrm{or}} \quad 2^{2m+1}(2k-1)+3, \end{aligned}$$ where $$k,m\in {\mathbb {N}} $$ , then there are no positive rational numbers x, y, z satisfying $$\begin{aligned} xyz = 1, \quad x+y+z = n. \end{aligned}$$ 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 $$\begin{aligned} xyz = a, \quad x+y+z = an. \end{aligned}$$ 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 holds: Then the system of equations $$\begin{aligned} xyz = a, \quad x+y+z = an. \end{aligned}$$ has no solutions in positive rational numbers x, y, z.