Sophie Germain Identity

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The Sophie Germain Identity states that:

$a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$

One can prove this identity simply by multiplying out the right side and verifying that it equals the left. To derive the factoring, first completing the square and then factor as a difference of squares:

\[a^4 + 4b^4 = a^4 + 4a^2b^2 + 4b^4 - 4a^2b^2 \\ \\ = (a^2 + 2b^2)^2 - (2ab)^2 \\ \\ = (a^2 + 2b^2 - 2ab) (a^2 + 2b^2 + 2ab)\]

Problems

Introductory

Intermediate

See Also

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