Fermat’s Last Theorem for Regular Primes

This FLT (for Regular Primes) is only first part of proof up till Ideal’s Ring Theory by Kumma.

The final complete proof by Andrew Wiles (1994) used more tools: Elliptical curve + Galois Theory.

The complete proof FLT by Andrew Wiles taking him 7 years in solitude, still a short time compared to 350 years before him but failed by the grandmasters Euler/Gauss etc. Today Andrew Wiles is hailed as the greatest 21CE Mathematician, even Fields Medal gave a Special Award to him (even he was > limit age of 40 years old ).

费马大定理 Fermat’s Last Theorem

费马大定理 Fermat’s Last Theorem (FLT): 17世纪业余数学家法国大法官费马开的一个”玩笑”, 推动350年来近代数学(Modern Mathematics)的突飞猛进。

1977秋 ~1979秋 笔者在法国-图卢斯(Toulouse, Southern France, Airbus 产地)费马学院 (College Fermat, aka Lycée Pierre de Fermat: Classe Préparatoire, 178th Batch)读两年的大学近代数学 (Mathématiques Supérieures et Spéciales), 尝过一生读书的”地狱”生活, 严谨(Mathematical Rigor)的思考训练, 像地鼠般(法国人戏称taupe)不见天日, 废寝忘食的煎熬。 当年对数学的恐惧, 终生牢牢铭记在心; 30年后”由惧转爱”, 数学竟然成为半退休后的业余嗜好, 享受数学的美 — 也是造物者宇宙天地的美!


FLT 350年数学长征英雄人物:
1. Fermat (费马 1601@ Toulouse, France)
2. Galois (伽罗瓦): Group Theory (群论)
3. Gauss (高斯)
4. Cauchy (柯西) Lamé (拉梅) Kummer (库马)
5. Solphie Germain
6. Euler (欧拉)
7. Taniyama (谷山丰), Shimura (志村五郎)

“数风流人物, 还看今朝”集大成者 :
8. Andrew Wiles (怀尔斯) 证明 (1994 -1995)”盒外思路” (Think Out of The Box): The Great Moment of 1994 Proof (YouTube)

\boxed {(1) = (2) = (3) }
(1). Elliptic Curve (椭圆曲线)
(2). Modular Form (模形式)
(3). Fermat’s Last Theorem (费马大定理)

费马大法官品尚清高, 讨厌政界官僚逢场作戏的应酬, 工余爱躲在家里玩数学, 然后写信和好友(巴斯卡 Pascal, 笛卡儿 Descartes,…)讨论, 无心中发明了物理(Optics)定律, 或然率 (Probability – 和Pascal合作), 解析几何 (Analytical Geometry – 和Descartes合作)…尤其他是近代数论(Number Theory)的开山鼻祖 (他的另一个Fermat’s Little Theorem今天用在电脑密码RSA Encryption)。
他偶然读到3,000年前希腊数学家Diaophantine的书 (10世纪阿拉伯人保存, 16世纪拉丁文翻译自阿拉伯文)。他心血来潮, 在书眉写道: “我找到一个漂亮的证明这题Diaophantine Equation, 但此书旁地方太小, 不能写下”。 他死后, 儿子整理遗作而发现此书, 就成为350年来的数学疑案。


Great Math Popular Books


日本数学科普作家写的好书。深入浅出, 适合中学生读最高深数学。

結城 浩 Hiroshi Yuki (1963 -) is a Japanese Math Popular Book Writer for Secondary and High School students. In the “Galois Theory” (Chapter 10) he boldly attempted to explain to them such complicated concepts: Quotient Group, Field Extension, Group Order, Normal Sub-Group, Solvable Group …

Evariste Galois‘s genius is he built a “bridge” between Field (域/体) and Group (群) – both new concepts invented by him. The “bridge” is called Galois Group, or by Emile Artin the “Group Automorphism”. He transferred the difficult problem of solving complicated 4 -ops (+-×/) Field (coefficients) to the single-op (permutation of roots) Group.

Galois Group is the ultimate TRUTH of all Math — Fermat’s Last Theorem, and any advanced Math, will use Galois Group or Field, to solve. Prof C.N. Yang 杨振宁 Nobel-Prize Physics discovery was based on Group Theory.

Evariste Galois was a French Math genius, died at 21 in a duel during French Revolution. He is the ‘Father’ of Modern Algebra. Failed 2 years in Ecole Polytechnique CONCOURS Entrance Exams, then kicked out by Ecole Normale Supérieure, his Math was not understood by all the 19th century World’s greatest Math Masters : Gauss, Fourier, Poisson, Cauchy…

“Galois Theory” — the ultimate Math “葵花宝典” (a.k.a. “Kongfu Bible“) — is only taught in the Math Honors Undergraduate or Masters degree Course.

自己学习《Galois Theory》(Page 365):
— “高中会教这种困难的数学吗 ?”
— “…我觉得比起給高中老师教, 不如自己好好学吧。”
— ” 重要的是自己学习。”




Our Daily Story #2: The man who cracked FLT

Follow up with the story #1 on FLT (Fermat’s Last Theorem),  it was finally cracked 358 years later in 1994 by a British mathematician Professor Andrew Wiles in Cambridge.
The proof of FLT is itself another exciting story, a 7-year lonely task on the attic top of his Cambridge house, nobody in the world knew anything about it, until the very day when Prof Wiles gave a seemingly unrelated lecture which ended with his announcement: FLT is finally proved. The whole world was shocked!


Part 1/5 Andrew Wiles and FLT Proof:

(Part 2 – 5 to follow from YouTube)

Speech at IMO by Andrew Wiles:

Modular Form

Modular Form (MF):
Is a function which takes Complex numbers from the upper half-plane as inputs and gives Complex numbers as outputs.

MF are notable for their high level of symmetry, determined not by a single number (2π for sine) but by 2×2 Matrices of Complex numbers.

1. Proof of the FLT
2. Investigation of Monster Group.
3. Elliptic curve = MF
4. L-function provides dictionary for translating between Analysis and Number Theory.