1983, en: gerd faltings proved Mo Deer's conjecture, and it was concluded that when N >: 2 (n is an integer), coprime A, B and C have only finite sets, which makes an+bn = cn.
1986, gerhardt Frey put forward "ε-conjecture": if a, b, c, a n+b n = c n, that is, if Fermat's last theorem is wrong, the elliptic curve y 2 = x (x-a n) (x+b). Frey's guess was immediately confirmed by Kenneth Rebett. This conjecture shows the close relationship between Fermat's Last Theorem and elliptic curve and module form.
1995 wiles and Taylor proved the Taniyama-Zhicun conjecture in a special case, and the Frey elliptic curve is just in this special case, thus proving Fermat's last theorem.
Wiles's process of proving Fermat's Last Theorem is also very dramatic. It took him seven years to obtain most of the evidence without being known. Then in June of 1993, he published his certificate at an academic conference, which immediately became the headlines of the world. But in the process of examining and approving the certificate, the experts found a very serious mistake. Wiles and Taylor then spent nearly a year trying to remedy it, and finally succeeded in a method abandoned by wiles in September 1994. This part of the proof is related to Iwasawa's theory. Their proof was published in 1995 Mathematics (en: Mathematics Yearbook).
1: Euler proved the case of n=3 by the unique factorization theorem.
2. Fermat himself proved that n=4.
3: 1825, Dirichlet and Legendre proved the case of n=5, using the extension of Euler method, but avoiding the unique factorization theorem.
4: 1839, French mathematician Lame proved the case of n=7. His proof uses a clever second tool closely combined with 7 itself, but it is difficult to generalize to the case of n= 1 1. Then, in 1847, he proposed the method of "cyclotomic integer" to prove it, but it didn't succeed.
5.kummer put forward the concept of "ideal number" in 1844, and he proved that Fermat's Last Theorem holds for all prime index n less than 100, and this research has reached a stage.
6. Leberg submitted a certificate, but it was rejected because there were loopholes.
7. Hilbert has also studied it, but there is no progress.
8: 1983 the german mathematician faltings proved an important conjecture-modal conjecture: the equation of square of x+square of y = 1 can be understood at most by a limited number. He won the Fields Medal for this.
At 9: 1955, Japanese mathematician Yutaka Taniyama first guessed that there was some connection between elliptic curve and another curve that mathematicians knew better-modulus curve. Gu Shan's conjecture was further refined by Wei Yi and tangcun Goro, forming the so-called "Gu Shan-tangcun conjecture". This conjecture shows that elliptic curves in rational number domain are all modular curves. This abstract conjecture confuses some scholars, but it makes the proof of Fermat's Last Theorem a step forward.
10: 1985, German mathematician Frey pointed out the relationship between "Gu Shan-intellectual village conjecture" and "Fermat's last theorem"; He put forward a proposition: if Fermat's last theorem is n > true, that is, there are a set of non-zero integers A, B and C, so that the n power of A+the n power of B = the n power of C (n >: 2), then the elliptic curve constructed by this set of numbers with the shape of Y squared = X (the n power of X +a) multiplied by (the n power of x-B) cannot be a modular curve. Despite his efforts, his proposition contradicts the "Gushan-Zhicun conjecture". If we can prove these two propositions at the same time, we can know that Fermat's Last Theorem is not based on reduction to absurdity, and this assumption is wrong, thus proving Fermat's Last Theorem. But at that time, he did not strictly prove his proposition.
1 1: 1986, American mathematician Bert proved Frey's proposition, so he wanted to focus on the "Gushan-Zhicun conjecture".
12:1In June, 1993, British mathematician Wells proved the "Taniyama-Zhicun conjecture" for a large class of elliptic curves in the rational number field. Because he showed in his report that Frey curve belongs to this elliptic curve, it also shows that he finally proved Fermat's last theorem. However, experts found loopholes in his proof, so Wells proved Fermat's Last Theorem completely and satisfactorily in September 1994 after more than a year's efforts.