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

TL;DR

This paper extends previous results on the non-existence of positive rational solutions for systems where the sum and product of three positive rationals are integers, under various modular and algebraic conditions.

Contribution

It generalizes earlier theorems by identifying broader conditions under which no positive rational solutions exist for the sum and product equations.

Findings

No solutions when n is divisible by 4 or congruent to 7 mod 8.

No solutions when a is divisible by 4 or 2 with n mod 4 equals 3.

Certain algebraic conditions involving powers of 2 and specific forms of a and n.

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 $x,y,z.$