Define a 2-Quadratic Group Operation as the following:
A 2nd degree polynomial of the form:
$$a_1x_1 + a_2x_2 + a_3x_1^2 + a_4x_2^2 + a_5x_1x_2 $$
Define a primal 2-quadratic group number as an integer $Q$ such that:
There do not exist integers $(x_1,x_2,x_3,x_4,x_5)$ such that:
$$a_1x_1 + a_2x_2 + a_3x_1^2 + a_4x_2^2 + a_5x_1x_2 = Q $$
The set of primal 2-quadratic group numbers for the 2-quadratic group operation $xy$ is trivially the set of prime numbers.
The concept outlined here generalizes the prime numbers.
It is well known that the $xy$ primality of a number can be tested in polynomial time. Is it true therefore that the generic 2-quadratic group primality of a number can be tested in polynomial time?
My argument would be no, but it is weak as it simply assumes that P != NP which is not yet known to be true.
Is there any other way to verify this?