# 费马大定理 Fermat’s Last Theorem

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)

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# Great Math Popular Books

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.

— “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。”

http://m.ruten.com.tw/goods/show.php?g=21437146332387

http://www.nh.com.tw/nh_bookView.jsp?cat_c=01&stk_c=9789866097010

https://tomcircle.wordpress.com/2014/03/21/math-girls-manga/

# 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!

http://en.m.wikipedia.org/wiki/Wiles%27_proof_of_Fermat%27s_Last_Theorem

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.

Uses:
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.

# Fermat Last Theorem

For all x, y, z integers,

$\mbox {FLT:} \: x^n + y^n = z^n$

$\mbox {If n} > 2 => \mbox {no solution for (x,y,z)}$

Proof:

Reduce n to 2 categories:

1. n |4:

Fermat proved n=4.

2. n not |4 => n|p, (p odd prime).

n=3 proved since 1770.

Conclude:

Prove FLT no solution for (n> 2) <=> Prove for  (odd primes p≥ 5)