Sophie Germain's identity
Sophie Germain's identity factors a⁴ + 4b⁴ as (a² − 2ab + 2b²)(a² + 2ab + 2b²), expressing a sum of squares as a product of quadratic polynomials.
Difference of squares: The identity x² − y² = (x − y)(x + y), factoring a difference of two squares. Writing a⁴ + 4b⁴ as (a² + 2b²)² − (2ab)² applies this identity directly.
Exponentiation: The operation of raising a quantity to a power by repeated multiplication. Recognizing fourth powers and squared quadratic expressions makes the factorization visible.
Fermat's Last Theorem: The theorem that xⁿ + yⁿ = zⁿ has no positive integer solutions for n greater than two. Sophie Germain used related factorization ideas in her work toward special cases of the theorem.
Sophie Germain: A French mathematician whose work contributed to number theory, elasticity, and mathematical physics. The identity is named after her, though its attribution and historical use require care.
Polynomial factorization: The expression of a polynomial as a product of simpler polynomials over a specified coefficient system. The identity is a specific factorization over the integers and reals.
Commutative ring: A ring whose multiplication is commutative. Polynomial identities such as this hold across commutative rings, though factor properties depend on the ring.
Sophie Germain's theorem: A theorem giving a criterion that can rule out certain solutions to Fermat's equation for a prime exponent. Its divisibility method is part of the same number-theoretic tradition as the identity.
Adrien-Marie Legendre: A French mathematician whose work included number theory, analysis, and geometry. Legendre developed results on Fermat’s equation connected with Germain’s number-theoretic work.
Sum of squares: An expression or number represented as the sum of two squared quantities. The quartic becomes factorable after it is rewritten as a difference of squares.
Integer: A whole number that is zero, positive, or negative. For integer a and b, the identity gives integer factors of a⁴ + 4b⁴.