* Fermat's principle

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 optics, Fermat's principle or the principle of least time is the principle 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 can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection.

Classically, Fermat's principle can be considered as a mathematical consequence of Huygens ' principle.

Fermat's principle is the main principle of quantum electrodynamics where it states that any particle ( e. g. a photon or an electron ) propagates over all available ( unobstructed ) paths and the interference ( sum, or superposition ) of its wavefunction over all those paths ( at the point of observer or detector ) gives the correct probability of detection of this particle ( at this point ).

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

... Fermat's principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.

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

Historically, Fermat's principle has served as a guiding principle in the formulation of physical laws with the use of variational calculus ( see Principle of least action ).

# REDIRECT Fermat's principle

Fermat's principle states that the path light takes between two points is the path that has the minimum optical path length.

* Fermat's principle

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

On the other hand, Fermat's last theorem has always been known by that name, even before it was proven ; it was never known as " Fermat's conjecture ".

Fermat's little theorem also has a nice generalization in finite fields.

In 1847 Gabriel Lamé announced a solution of Fermat's Last Theorem for all -- i. e., that the Fermat equation has no solutions in nonzero integers, but it turned out that his solution hinged on the assumption that the cyclotomic ring is a UFD.

A sample application of Faltings ' theorem is to a weak form of Fermat's Last Theorem: for any fixed n > 4 there are at most finitely many primitive integer solutions to a < sup > n </ sup > + b < sup > n </ sup > = c < sup > n </ sup >, since for such n the curve x < sup > n </ sup > + y < sup > n </ sup > = 1 has genus greater than 1.

Modulo p, Fermat's little theorem says it also has the same roots, 1, 2, ...,.

The speculative fiction writer, Ted Chiang, has a story, Story of Your Life, that contains visual depictions of Fermat's Principle along with a discussion of its teleological dimension.

In other ways it is the least accessible discipline ; for example, Wiles ' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in physics or the general public discourse.

When the demon has been banished Sam ends up playing with the Fermat's Theorem disproof fragment the demon left behind and thinking about summoning the demon and tricking him again.

But by Fermat's last theorem it is now known that ( for n ≥ 3 ) there are no nontrivial integer solutions to the Fermat equation ; therefore, the Fermat curve has no nontrivial rational points.

Fermat's theorem now has a lengthy and extremely technical orthodox proof drawing on many different areas of advanced mathematics.

Proof of ( 1 ) and ( 2 ): One has from Fermat's little theorem,

If p-1 does not divide 2n then after Fermat's theorem one has,

The room featured in Fermat's Room has a design similar to that of a hydraulic press.

It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, scrawled in the margins of Fermat's copy of ' Arithmetica ', which states that the equation, where,, and are non-zero integers, has no solution with greater than 2.

When they are both large, for instance more than 2000 bits long, randomly chosen, and about the same size ( but not too close, e. g. to avoid efficient factorization by Fermat's factorization method ), even the fastest prime factorization algorithms on the fastest computers can take enough time to make the search impractical ; that is, as the number of digits of the primes being factored increases, the number of operations required to perform the factorization on any computer increases drastically.

One obtains the value f ( r ) by substitution of the value r for the symbol X in P. One reason to distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function ( see Fermat's little theorem for an example where R is the integers modulo p ).

At the same time, Kummer developed powerful new methods to prove Fermat's Last Theorem at least for a large class of prime exponents using what we now recognize as the fact that the ring is a Dedekind domain.

At the same time they should not be too close together, or else the number can be quickly factored by Fermat's factorization method.

Neither are prime numbers of the form because Fermat's theorem on sums of two squares assures us they can be written for integers and, and.

It is a form of Fermat's spiral.

Amongst his most original contributions were: his " Conjecture II " ( still open ) on Galois cohomology ; his use of group actions on Trees ( with H. Bass ); the Borel-Serre compactification ; results on the number of points of curves over finite fields ; Galois representations in ℓ-adic cohomology and the proof that these representations have often a " large " image ; the concept of p-adic modular form ; and the Serre conjecture ( now a theorem ) on mod-p representations that made Fermat's last theorem a connected part of mainstream arithmetic geometry.

One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form, where x, y are integers.

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 )

In the integer case this identity finds applications in number theory for example when used in conjunction with one of Fermat's theorems it proves that the product of a square and any number of primes of the form 4n + 1 is also a sum of two squares.

The structural result of Mordell, that the rational points on an elliptic curve E form a finitely-generated abelian group, used an infinite descent argument based on E / 2E in Fermat's style.

Fermat's theorem on sums of two squares is then equivalent to the statement that a prime is represented by the form ( i. e.,, ) exactly when is congruent to modulo.

Thus, to prove Fermat's theorem it is enough to find any reduced form of discriminant − 1 that represents.

Dedekind gave at least two proofs of Fermat's theorem on sums of two squares, both using the arithmetical properties of the Gaussian integers, which are numbers of the form a + bi, where a and b are integers, and i is the square root of − 1.

In its simplest form, Fermat's method might be even slower than trial division ( worst case ).

When does not lie in the subset, this polynomial is irreducible in K, and its splitting field over K is a cyclic extension of K of degree p. This follows since for any root β, the numbers β + i, for, form all the roots — by Fermat's little theorem — so the splitting field is.