It is well known that If a prime is of the form $x^2 + y^2$ where $x,y$ are positive integers , then there is only 1 such representation for that prime.
This uniqueness seems to be rare.
For instance the cube analog fails :
$$ 10^3 + 9^3 = 1^3 + 12^3 = 1729 $$
Since $1729$ is a prime.
It Also fails for triangular numbers
$$31 = 28 + 3 = 21 + 10 $$
Now I am aware of the connections to norms of a UFD. For instance $x^2 + y^2 $ is the norm for the gaussian integers ( ring $Z(\sqrt -1)$ ).
But most integer-valued polynomials are not norms of rings.
So are there examples of this Uniqueness for primes of an integer-valued polynomial that is not the norm of a ring ?
Notice I do not accept $u^4 + v^4$ as an answer because that is Also of the form $x^2 + y^2$.
I would like to point out that polynomials with non-negative rational coëfficients are usually not norms ! In fact they are not When the Number of variables or the degree is above $2$.
I wonder If we have uniqueness for every prime $p$ of the form
$$ p = a^6 + b^6 + c^6 $$
Everytime I think or read about number-theory or ring theory this idea bothers me !
Integer-valued polynomial means a polynomial with rational coëfficiënts that maps every integer to an integer. Example triangular numbers from $D(D+1)/2$.