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-1quad { m or} quad 2^{2m+1}(2k-1)+3,

where

k, m i n m a t h b b 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 i n m a t h b b N

be odd and let either

n e q u i v 0 p m o d 4

or

n e q u i v 7 p m o d 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 i n m a t h b b N

are such that at least one of the following conditions hold:

n e q u i v 0 p m o d 4

n e q u i v 7 p m o d 8

a e q u i v 0 p m o d 4

a e q u i v 0 p m o d 2

and

n e q u i v 3 p m o d 4

a^{2} n^{3} = 2^{2 m + 1} (2 k - 1) + 27

for some

k, m i n m a t h b b N .

Then the system of equations

xyz = a, quad x+y+z = an.

has no solutions in positive rational numbers

x, y, z .