Fermat’s Last Theorem

Everyone hs thought of it as the dream for mathmatics to solve such problem. – Will Hearst at Palace of Fine Arts San Francisco, July 28th, 1993.

The 350-year old problem was solved by Andrew Wiles in 1993. The story starts with Fermat writing on the margin of Diophantus’ Arithmetica. He wrote like this:

Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet (Nagell 1951, p. 252)

Which roughly translates to:

It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

Thus, the notorious problem that persisted for 350 years.

Analytical Geometry의 탄생.

1601년 8월 17일 프랑스의 변호사이자 수학자인 Pierre de Fermat이 태어났다. “페르마의 마지막 정리”라는 358년간 수학자들을 괴롭힌 수학 문제를 만든 사람이자 빛이 퍼지는 법칙을 연구한 사람이기도 하다.

Euclid와 Elements Al Quarzmi의 Algebra의 결과를 합친 다음과 같은 결과를 얻을 수 있었다.

\[z = \sqrt[3]{q + \sqrt{q^2-p^3}} + \sqrt[3]{q-\sqrt{q^2-p^3}}\]

이것은 $ x^3-3px-2q $의 근이다.

The Proof of Fermat’s Last Theorem

• Suppose $a^n + b^n = c^n$
• Write down $y^2 = x(x-a^n)(x+b^n).$
• This is a semistable elliptic curve.
• Wiles’ 1993 theorem says it’s modular
• Ribet’s 1986 theorem says it can’t be modular. (Uses that $a^n + b^n$ is $c^n.$)
• This contradiction shows that there are no a, b, and c with $a^n + b^n = c^n$.

History for Fermat’s Last Theorem Solution.

• Y. Taniyama, 1955. Conjecture: elliptic curves are modular.
• G. Frey, 1985. Fermat counterexample leads to elliptic curve. This curve seems non-modular.
• J-P. Serre, 1985. To show Frey’s curve is non-modular, it suffices to prove two facts about modular forms.
• K. Ribet, 1986. Frey’s curve is non-modular.
• A. Wiles, 1993. Frey’s curve is modular.

Modular elliptic curves

An elliptic curve can be modular.

Instead of asking how often

\[y^2 = x^3 - x \quad \text{(i.e., } y^2-(x^3-x) = 0,\]

we will ask, for every prime number $p$: how often is $y^2-(x^3-x)$ a multiple of $p$?

\[\text{Example: } p = 5 \\ x\\ y \begin{array}{c | c c c c c } & 0 & 1 & 2 & 3 & 4 \\ \hline 0& \boxed{0} & \boxed{0} & -6&-24&\boxed{-60}\\ 1& 1 & 1 & \boxed{-5}&-23&-59\\ 2& 4 & 4 & -2&\boxed{-20}&-56\\ 3& 9 & 9 & 3 &\boxed{-15}&-51\\ 4&16 &16 &\boxed{-10}& -8&-44\\ \end{array}\] \[y^2-(x^3-x) \equiv 0\ (\textrm{mod}\ p)\]

Counting gives these tables:

\[\begin{array}{c|c c c c c c c c c c c} p & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 \\ \hline N_p & 2 & 3 & 7 & 7 & 11 & 7 & 15 & 19 & 23 & 39 & 31 \\ \end{array}\] \[\begin{array}{c|c c c c} p & 1000003 & 1000033 & 1000037 & 1000039 \\ \hline N_p & 1000003 & 998207 & 998055 & 1000039 \\ \end{array}\]

In 1814, Gauss found a recipe for calculating $N_p$ for this elliptic curve:

• $N_2 = 2$
• If $p$ is $1$ less than a multiple of $4$, then $N_p=p$.
• If $p$ is $1$ more than a multiple of $4$, then there is a more complicated formula.

A Sequence must be very special to have this modular property.

Conjecture (Taniyama 1955, Shimura 1962).

Every elliptic curve is modular