So I was consider Lagrange's 4-square theorem and came up with this generalization:
Given a polynomial with rational coefficients
$$P(x) = a_0 + a_1x + ... + a_nx^n$$
Determine if there exists numbers $N$ and $W$ and find the smallest $N$ such that all integers greater than $W$ can be expressed as
$$P(x_1) + P(x_2)+ ... + P(x_N)$$
Naturally the case of
$$P(x) = x^2$$ Leads itself to $N = 4$ $W = -1$ as proven by lagrange
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It appears that a method of proof could be sketeched as follows:
determine $f(x,y)$ such that $P(f(x,y)) = f(P(x),P(y))$
Note that every integer must have a representation of the form $f(w_1, f(w_2, f(w_3 ... (f(w_{n-1}, w_n))...))) $
Now generate a set of primal elements of such that every integer can be formed using compositions of $f$ with these elements as arguments $f(x,y) = xy$ has the Primes as its primal elements.
Now prove each of these primal elements has a representation using the polynomial $P(x)$