KnowraSophie Germain's theoremSophie Germain's theoremFor an odd prime p such that 2p+1 is prime, Fermat’s equation x^p + y^p = z^p has no nonzero integer solutions.BriefConnectFermat's Last Theorem: The claim that xⁿ + yⁿ = zⁿ has no positive integer solutions for every integer n greater than 2. Sophie Germain’s result proves the theorem for a specified class of prime exponents.Prime number: An integer greater than one whose only positive divisors are one and itself. Both p and 2p+1 must be prime for the theorem’s stated conclusion.Sophie Germain: A French mathematician whose work included major contributions to number theory and elasticity. She proved the prime-exponent result that bears her name.Fermat's Last Theorem for exponent 3: The proof that x³ + y³ = z³ has no positive integer solutions. Exponent 3 is not covered because 2·3+1 is composite.Fermat's equation: The Diophantine equation xⁿ + yⁿ = zⁿ, central to Fermat’s Last Theorem. The theorem rules out its nonzero integer solutions when the exponent meets its prime conditions.Sophie Germain prime: A prime p for which 2p+1 is also prime. These are exactly the exponents covered by the theorem.Fermat's Last Theorem in the 19th century: Nineteenth-century efforts to prove Fermat’s Last Theorem using arithmetic and algebraic number theory. Germain’s result was one of the strongest general advances in those efforts.Fermat's Last Theorem for exponent 7: The proof that x⁷ + y⁷ = z⁷ has no positive integer solutions. Exponent 7 is not covered because 2·7+1 is composite.Cyclotomic polynomial: A polynomial whose roots are the primitive roots of unity of a given order. Factoring x^p + y^p separates the sum into factors whose divisibility drives the argument.Diophantine equation: An equation whose solutions are sought among integers or other specified rational numbers. The theorem excludes integer solutions to a particular Diophantine equation.Show all 20Linked from 1 pageSophie Germain's identityRelated: Its divisibility method is part of the same number-theoretic tradition as the identity.