Sophie Germain's Identity

Theorem

For any two numbers $x$ and $y$:

$x^4 + 4 y^4 = \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}$

Proof 1

\(\ds \)

\(\)

\(\ds \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}\)

\(\ds \)

\(=\)

\(\ds x^4 + x^2 \cdot 2 y^2 - x^2 \cdot 2 x y + x^2 \cdot 2 y^2 + 4 y^2 - 2 y^2 \cdot 2 x y + x^2 \cdot 2 x y + 2 y^2 \cdot 2 x y - 2 x y \cdot 2 x y\)

\(\ds \)

\(=\)

\(\ds x^4 + 4 y^4\)

gathering up terms and cancelling

$\blacksquare$

Proof 2

\(\ds \)

\(\)

\(\ds \paren {x^2 + 2 y^2 + 2 x y} \paren {x^2 + 2 y^2 - 2 x y}\)

\(\ds \)

\(=\)

\(\ds \paren {x^2 + 2 y^2}^2 - \paren {2 x y}^2\)

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds x^4 + 2 \cdot x^2 \cdot 2 y^2 + 4 y^2 - 2 x y \cdot 2 x y\)

\(\ds \)

\(=\)

\(\ds x^4 + 4 y^4\)

gathering up terms and cancelling

$\blacksquare$

Source of Name

This entry was named for Marie-Sophie Germain.

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.19$
• 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 2$: Special Products and Factors: $2.19.$