<s>[INST] Reducible polynomials similar to Sophie Germain's identity Are there known any other simple nontrivial polynomials in two or more variables that can be factored like the one in Sophie Germain's identity?
$$x^4+4y^4=(x^2-2 x y+2 y^2) (x^2+2 x y+2 y^2)$$
Besides the well known (for $n \in \mathbb {Z} ^{+}$):
$$x^n-y^n=(x-y) \sum _{i=1}^n x^{i-1} y^{n-i}$$
$$x^{2 n-1}+y^{2 n-1}=(x+y) \sum _{i=1}^{2 n-1} (-1)^{i-1} x^{i-1} y^{2 n-1-i}$$
And of course besides trivial ones like $5 x^2+x y^2=x (5 x+y^2)$, where all terms share a common factor.
By simple polynomial I mean a polynomial that has few terms. The less terms the better - ideally only two terms like in the Germain's polynomial or in the known ones.
 [/INST] More formulas similar to Sophie Germain's identity can be derived from Aurifeuillean factorizations such as
$$x^6+27y^6=(x^2+3y^2)(x^2-3xy+3y^2)(x^2+3xy+3y^2),$$
$$x^{10}-3125y^{10}=(x^2-5y^2)(x^4-5x^3 y+15x^2 y^2-25x y^3+25y^4)(x^4+5x^3 y+15x^2 y^2+25x y^3+25y^4).$$
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