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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 $P(x)$

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    $\begingroup$ For $x^m$, please see Waring's Problem. There has been work with other polynomials, for example $x(x-1)/2$ is interesting. $\endgroup$ – André Nicolas Jul 22 '14 at 14:48
  • $\begingroup$ Googling generalizations of Waring's problem will yield a fair number of hits. $\endgroup$ – André Nicolas Jul 22 '14 at 15:09

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