Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal.
To be precise: Let $x<y<z$ and $x'<y'<z'$ be positive integers such that $x+y+z=x'+y'+z'$ and $xyz=x'y'z'$.
Is it true that the maximum of these numbers, i.e. $\max(z,z')$, cannot be a prime number?
Experiments up to $\max(z,z')=43$ confirm this, but I didn't give it any more thought.
First point is that you can't have $z=z'$. To see that, note that equality of the $z's$ would imply that $$x+y=x'+y'\;\;\&\;\;xy=x'y'$$ But that would imply $x=x',y=y'$ (if $(x+y)=A$ and $xy=B$ with $y>x$ then $y-x=\sqrt{A^2-4AB}$ so you can solve for $x,y$.)
Now suppose that $z=p$ was the maximum. Then $p>z'>y'>x'$ but $p\,|\,x'y'z'$, an impossibility if $p$ is prime.