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In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) 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. [1]
Ribet, Ken (1995). "Galois representations and modular forms" (PDF) . Discusses various material which is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama–Shimura.
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.
His article was published in 1990. In doing so, Ribet finally proved the link between the two theorems by confirming, as Frey had suggested, that a proof of the Taniyama–Shimura–Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem.
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]
Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT).
In the summer of 1986, Ken Ribet demonstrated that, just as Gerhard Frey had anticipated, a special case of the Taniyama–Shimura conjecture (still not proved at the time), together with the now proved epsilon conjecture (now called Ribet's theorem), implies Fermat's Last Theorem.
In 1986, Ken Ribet proved that if the Taniyama–Shimura conjecture held, then so would Fermat's Last Theorem, which inspired Andrew Wiles to work for a number of years in secrecy on it, and to prove enough of it to prove Fermat's Last Theorem. Owing to the pioneering contribution of Wiles and the efforts of a number of mathematicians, the ...