He is most famous for proving Fermat's Last Theorem.

Wiles discovered Fermat's Last Theorem on his way home from school when he was 10 years old.

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

As an unproven conjecture that eluded brilliant mathematicians ' attempts to either prove it or disprove it for generations, his statement became famous as Fermat's Last Theorem.

The most celebrated single question in the field, the conjecture known as Fermat's Last Theorem, was solved by Andrew Wiles but using tools from algebraic geometry developed during the last century rather than within number theory where the conjecture was originally formulated.

concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found " a truly marvelous proof of this proposition ," now referred to as Fermat's Last Theorem.

Problem II. 8 in the Arithmetica ( edition of 1670 ), annotated with Fermat's comment which became Fermat's Last Theorem.

Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations — including the " Last Theorem "— were printed in this version.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

While in Copenhagen, Abel did some work on Fermat's Last Theorem.

Early attempts to prove Fermat's Last Theorem climaxed when Kummer introduced regular primes, primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions

Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after.

Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it très ingénieuse ( See Best Work on Fermat's Last Theorem ).

The first letter, dated 21 November 1804, discussed Gauss ' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem.

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem.

In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem.

She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case ( see Best Work on Fermat's Last Theorem ).

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

By Fermat's Little Theorem, 2 < sup >( q − 1 )</ sup > ≡ 1 ( mod q ).

Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that ( q − 1 )< sup >( p − 1 )</ sup > ≡ 1 ( mod p ).

* Ideal Numbers, Proof that the theory of ideal numbers saves unique factorization for cyclotomic integers at Fermat's Last Theorem Blog.

* Ferdinand Eisenstein by Larry Freeman ( 2005 ), Fermat's Last Theorem Blog.

His written works include Fermat's Last Theorem ( in the United States titled Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem ), The Code Book ( about cryptography and its history ), Big Bang ( about the Big Bang theory and the origins of the universe ) and Trick or Treatment?

The laws of reflection and refraction can be derived from Fermat's principle which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Fermat's principle leads to Snell's law ; when the sines of the angles in the different media are in the same proportion as the propagation velocities, the time to get from P to Q is minimized.

In the classic mechanics of waves, Fermat's principle follows from the extremum principle of mechanics ( see variational principle ).

Indeed Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as Huygens ' principle.

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time.

In number theory, the Fermat pseudoprimes make up the most important class of pseudoprimes that come from Fermat's little theorem.

It is called the " little theorem " to distinguish it from Fermat's last theorem.

The play opens on 10 April 1809, in a garden front room of a country house in Derbyshire with tutor Septimus Hodge trying to distract his 13 year-old pupil Thomasina from her enquiries as to the meaning of a " carnal embrace " by challenging her to prove Fermat's Last Theorem so he can focus on reading the poem ' The Couch of Eros ', a piece written by another character, Mr. Ezra Chater.

Fermat's test for compositeness, which is based on Fermat's little theorem, works as follows: given an integer n, choose some integer a coprime to n and calculate a < sup > n − 1 </ sup > modulo n. If the result is different from 1, n is composite.

, with some help from Richard Taylor, proved the Taniyama – Shimura – Weil conjecture for all semistable elliptic curves, which was strong enough to yield a proof of Fermat's Last Theorem.

Several theorems in number theory similar to Fermat's Last Theorem follow from the modularity theorem.

The motivation for this definition is the fact that all prime numbers p satisfy the above equation which can be deduced from Fermat's little theorem.

Humboldt also secured a recommendation letter from Gauss, who upon reading his memoir on Fermat's theorem wrote with an unusual amount of praise that " Dirichlet showed excellent talent ".

A Cullen number C < sub > n </ sub > is divisible by p = 2n − 1 if p is a prime number of the form 8k-3 ; furthermore, it follows from Fermat's little theorem that if p is an odd prime, then p divides C < sub > m ( k )</ sub > for each m ( k ) = ( 2 < sup > k </ sup > − k )

It is widely believed that Kummer was led to his " ideal complex numbers " by his interest in Fermat's Last Theorem ; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Dirichlet told him his argument relied on unique factorization ; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.

A slightly more rigorous definition of a light ray follows from Fermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Isaac Newton would later write that his own early ideas about calculus came directly from " Fermat's way of drawing tangents.