Sophie Germain's identity

In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that

{\begin{aligned}x^{4}+4y^{4}&={\bigl (}(x+y)^{2}+y^{2}{\bigr )}\cdot {\bigl (}(x-y)^{2}+y^{2}{\bigr )}\\&=(x^{2}+2xy+2y^{2})\cdot (x^{2}-2xy+2y^{2}).\end{aligned}}

Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form

x^{4}+4y^{4}

, and it frequently forms the basis of problems in mathematics competitions.^[1]^[2]^[3]

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