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.