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Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.
In the generalized Fermat’s principle [6] the time is used as a functional and together as a variable. It is applied Pontryagin’s minimum principle of the optimal control theory and obtained an effective Hamiltonian for the light-like particle motion in a curved spacetime. It is shown that obtained curves are null geodesics.
The general results presented above for Hamilton's principle can be applied to optics using the Lagrangian defined in Fermat's principle.The Euler-Lagrange equations with parameter σ =x 3 and N=2 applied to Fermat's principle result in ˙ = with k = 1, 2 and where L is the optical Lagrangian and ˙ = /.
Kenneth Alan Ribet (/ ˈ r ɪ b ɪ t /; born June 28, 1948) is an American mathematician working in algebraic number theory and algebraic geometry.He is known for the Herbrand–Ribet theorem and Ribet's theorem, which were key ingredients in the proof of Fermat's Last Theorem, as well as for his service as President of the American Mathematical Society from 2017 to 2019.
The most spectacular application of the conjecture is the proof of Fermat's Last Theorem (FLT). Suppose that for a prime p ≥ 5, the Fermat equation + = has a solution with non-zero integers, hence a counter-example to FLT. Then as Yves Hellegouarch was the first to notice, [19] the elliptic curve
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory.He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Knight Commander of the Order of the British Empire in 2000. [1]
For example, if a = 2 and p = 7, then 2 7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] ().
Gerhard Frey (German:; born 1 June 1944) is a German mathematician, known for his work in number theory.Following an original idea of Hellegouarch, [1] he developed the notion of Frey–Hellegouarch curves, a construction of an elliptic curve from a purported solution to the Fermat equation, that is central to Wiles's proof of Fermat's Last Theorem.