Sophie Germain's theorem
Encyclopedia
In number theory
, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation xp + yp = zp of Fermat's Last Theorem
. Specifically, Sophie Germain
proved that the product xyz must be divisible by p2 if an auxiliary prime θ can be found such that two conditions are satisfied:
Conversely, the first case of Fermat's Last Theorem must hold for every prime p for which even one auxiliary prime can be found. Germain identified such an auxiliary prime θ for every prime less than 100. The theorem and its application to primes p less than 100 were attributed to Germain by Adrien-Marie Legendre
in 1823.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation xp + yp = zp of Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. Specifically, Sophie Germain
Sophie Germain
Marie-Sophie Germain was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a gender-biased society, she gained education from books in her father's library and from correspondence with famous mathematicians such as...
proved that the product xyz must be divisible by p2 if an auxiliary prime θ can be found such that two conditions are satisfied:
- No two pth powers differ by one moduloModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
θ; and - p is itself not a pth power moduloModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
θ.
Conversely, the first case of Fermat's Last Theorem must hold for every prime p for which even one auxiliary prime can be found. Germain identified such an auxiliary prime θ for every prime less than 100. The theorem and its application to primes p less than 100 were attributed to Germain by Adrien-Marie Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
in 1823.