Sophie Germain's theorem

Last updated December 15, 2020

In number theory, Sophie Germain's theorem is a statement about the divisibility of solutions to the equation $x^{p}+y^{p}=z^{p}$ of Fermat's Last Theorem for odd prime $p$ .

Formal statement

Specifically, Sophie Germain proved that at least one of the numbers $x$ , $y$ , $z$ must be divisible by $p^{2}$ if an auxiliary prime $q$ can be found such that two conditions are satisfied:

No two nonzero $p^{\mathrm {th} }$ powers differ by one modulo $q$ ; and
$p$ is itself not a $p^{\mathrm {th} }$ power modulo $q$ .

Conversely, the first case of Fermat's Last Theorem (the case in which $p$ does not divide $xyz$ ) must hold for every prime $p$ for which even one auxiliary prime can be found.

History

Germain identified such an auxiliary prime $q$ 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.^[1]

Notes

↑ Legendre AM (1823). "Recherches sur quelques objets d'analyse indéterminée et particulièrement sur le théorème de Fermat". Mém. Acad. Roy. des Sciences de l'Institut de France. 6. Didot, Paris, 1827. Also appeared as Second Supplément (1825) to Essai sur la théorie des nombres, 2nd edn., Paris, 1808; also reprinted in Sphinx-Oedipe4 (1909), 97–128.

Related Research Articles

Quadratic reciprocity Gives conditions for the solvability of quadratic equations modulo prime numbers

In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is:

Sophie Germain French mathematician, physicist, and philosopher

Marie-Sophie Germain was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Leonhard Euler, and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of prejudice against her sex, she was unable to make a career out of mathematics, but she worked independently throughout her life. Before her death, Gauss had recommended that she be awarded an honorary degree, but that never occurred. On 27 June 1831, she died from breast cancer. At the centenary of her life, a street and a girls’ school were named after her. The Academy of Sciences established the Sophie Germain Prize in her honor.

Lagrange's theorem, in group theory, a part of mathematics, states that if $H$ is a subgroup of a finite group $G$ , then the order of $H$ divides the order of $G$ . The theorem is named after Joseph-Louis Lagrange. The following variant also identifies the ratio $, as being the index [G : H], defined as the number of left cosets of H in G .$

Adrien-Marie Legendre French mathematician

Adrien-Marie Legendre was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.

Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$ , the number $a p - a$ is an integer multiple of $p$ . In the notation of modular arithmetic, this is expressed as

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.

This article collects together a variety of proofs of Fermat's little theorem, which states that

In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d. The numbers of the form a + nd form an arithmetic progression

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:

In number theory, a Wieferich prime is a prime number p such that p² divides $2 p - 1 - 1$ , therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides $2 p - 1 - 1$ . Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's last theorem, at which time both of Fermat's theorems were already well known to mathematicians.

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. That is, the squares form an additive basis of order four.

In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

Fermat's theorem on sums of two squares asserts that an odd prime number p can be expressed as

In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre, states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.

Fermats Last Theorem Famous 17th century conjecture proved by Andrew Wiles in 1994

In number theory, Fermat's Last Theorem states that no three positive integers $a$ , $b$ , and $c$ satisfy the equation $a n + b n = c n$ for any integer value of $n$ greater than 2. The cases $n = 1$ and $n = 2$ have been known since antiquity to have infinitely many solutions.

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n larger than 2. In the centuries following the initial statement of the result and before its general proof, various proofs were devised for particular values of the exponent n. Several of these proofs are described below, including Fermat's proof in the case n = 4, which is an early example of the method of infinite descent.

In mathematics, elliptic curve primality testing techniques, or elliptic curve primality proving (ECPP), are among the quickest and most widely used methods in primality proving. It is an idea put forward by Shafi Goldwasser and Joe Kilian in 1986 and turned into an algorithm by A. O. L. Atkin the same year. The algorithm was altered and improved by several collaborators subsequently, and notably by Atkin and François Morain, in 1993. The concept of using elliptic curves in factorization had been developed by H. W. Lenstra in 1985, and the implications for its use in primality testing followed quickly.

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to $Q$ , the field of rational numbers. The $n$ -th cyclotomic field $Q (ζ n)$ is obtained by adjoining a primitive $n$ -th root of unity $ζ n$ to the rational numbers.

In mathematics, Legendre's three-square theorem states that a natural number can be represented as the sum of three squares of integers

References

Laubenbacher R, Pengelley D (2007) "Voici ce que j'ai trouvé": Sophie Germain's grand plan to prove Fermat's Last Theorem
Mordell LJ (1921). Three Lectures on Fermat's Last Theorem. Cambridge: Cambridge University Press. pp. 27–31.
Ribenboim P (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer-Verlag. pp. 54–63. ISBN 978-0-387-90432-0.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[1] Legendre AM (1823). "Recherches sur quelques objets d'analyse indéterminée et particulièrement sur le théorème de Fermat". Mém. Acad. Roy. des Sciences de l'Institut de France. 6. Didot, Paris, 1827. Also appeared as Second Supplément (1825) to Essai sur la théorie des nombres, 2nd edn., Paris, 1808; also reprinted in Sphinx-Oedipe4 (1909), 97–128.

[1]