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。


Our Daily Story #5: The Prince of Math

Carl Friedrich Gauss is named the “Prince of Math” for his great contributions in almost every branch of Math.

As a child of a bricklayer father, Gauss used to follow his father to construction site to help counting the bricks. He learned how to stack the bricks in a pile of ten, add them up to obtain the total. If a pile has only 3, for example, he would top up 7 to make it 10 in a pile. Then 15 piles of 10 bricks would give a total of 150 bricks.

One day in school, his teacher wanted to occupy the 9-year-old children from talking in class, made them add the sum:
1 + 2 + 3+ ….+ 98 + 99 + 100 = ?

Gauss was the first child to submit the sum within few seconds = 5,050.

He used his brick piling technique: add

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

49 + 52 = 101
50 + 51 = 101

In the 100 numbers broken down in pair of two as above, there are 50 pairs of sum 101, which gives total of 5050.
This is a high school math formula for the Sum (S) of a series: {(1, 2, 3,... n)}
\boxed {S = \frac {n.(n+1)} {2} }




Gauss Library Records

If we were to choose only 3 greatest scientists in the entire human history, who excelled in every field of science and mathematics, they are:
1) Archimedes
2) Issac Newton
3) Carl Friedrich Gauss

Let’s see how Gauss became a great scientist in his formative years in the university, it would give us a clue by knowing what kind of books did he read ?

Carl Friedrich Gauss was awarded a 3-year ‘overseas’ scholarship to study in Göttingen University (located in the neighboring state Hanover) by his own state sponsor the Duke of Brunswick.

Gauss chose Göttingen University because of its rich collection of books.
During the 3 years, he read very widely on average 8 books in a month.

Below was his student days’ library records:

1795-1796 (1st semister): total 35 books
Math (M) :1 ,
Astrology (A):2,
History/Philosophy (H): 1,
Literature/ Language (L): 15,
Science Journal (S): 16

1796-1797 (1st semister): total 47
General Science: 3,
Math (M) :6,
Astrology (A):9,
Mechanics / Physics (P) : 1,
Literature/ Language (L): 4,
Science Journal (S): 23,
Others: 1

1796-1797 (2nd semister): total 35
General Science: 3,
Math (M) :6,
Astrology (A):1,
Mechanics / Physics (P): 3,
History/Philosophy: 1
Literature/ Language (L): 5,
Music: 1
Science Journal (S): 13
Others: 2

1797-1798 (1st semister): total 52
Math (M) :3,
Astrology (A):1,
Mechanics / Physics (P): 1,
History/Philosophy: 1
Literature/ Language (L): 1,
Music: 3
Science Journal (S): 41
Others: 1

1797-1798 (2nd semister): total 24
Math (M) :1,
Astrology (A):2,
Mechanics / Physics (P): 1,
History/Philosophy: 3,
Literature/ Language (L): 4,
Science Journal (S): 13,
Others: 2

Grand total (3 years): 193 books

Gauss read directly the great masters’ classics from Euler (6 books), Newton, Fermat etc, and Mechanics from Lagrange…

During these fruitful years, Gauss made major mathematics discoveries:
– the ancient Greek “impossible” problem of the construction of heptadecagon (17-sided polygon) with only compass and (unmarked) ruler;
– Modular arithmetic;
– Quadratic Reciprocity Law;
– at 21 wrote the Number Theory masterpiece “Disquisitiones Arithmeticae” (Latin, Arithmetical Investigations).

Nephew and Maternal Uncle

There is a common proverb in my Chinese dialect Fujian spoken today in China Fujian province, Taiwan, Singapore and Malaysia, which says
“A nephew is like his maternal uncle”  外甥像母舅
In modern Biology we know mother passes some genes to her children. Some disease like colorblind is carried by mother down to her sons, the mother herself is immune but her brothers are colorblind as the nephews.
Interesting behavior, intelligence are also similarly inherited from mother and maternal uncles.

Two greatest mathematicians in the history, Newton and Gauss, were the lucky nephews from their maternal uncles who were highly educated to spot the nephew’s genius, although the boys’ parents were uneducated.

Newton’s father died early, mother Hannah Ayscough had a brother William Ayscough educated in Cambridge. William convinced Hannah to send the talented boy Newton to Cambridge.

Gauss’s father was a bricklayer, mother Dorothy Benz had a younger brother Friedrich Benz, who was an intelligent man, rescued young Gauss from following his father’s footsteps as bricklayer.

I met two talented boys, one won the top International Math Olympiad prize, the other excels in the national PSLE exams, their respective mother and father are not highly educated, but their maternal uncles are top scholars.

However, there are serious consequences of (Parallel or Cross) cousin’s inter-marriage from maternal link. Charles Darwin first discovered this genetic problem because he and his wife were such cousins.

The ancient Chinese liked to marry between cousins from maternal side. We praised the practice as “close relatives getting closer” 亲上加亲. The daughter-in-law would likely get along well with the mother-in-law because of the maternal family blood link. The last Qing dynasty emperors were the products of such marriage custom, and most of them were incompetent emperors who died young.

This widely mal-practice of inter-marriage had probably been the cause of the decline of the brilliant Chinese civilization from 14th century (Ming dynasty). China went to sleep for 500 years till awaken from 1911 – as Napoleon imprisoned by the British in St. Helena Island warned the British not to wake up the ‘sleeping lion’, else China would shake the world (*) (she does today as the World’s 2nd super-power) – by the European colonizers (#) in Opium wars and the Japanese invasion.

This Antropology of Kinship Problem is interestingly a modern math problem in Group Theory, first studied by André Weil for the Australian aborigines.

Notes (*): Napoleon’s famous warning to his British enemy before the Opium Wars which humiliated China:
“Quand la Chine s’eveillera, la terre tremblera.”

(#): England, France, Germany, Russia, Japan, USA, Austria and Italy. Ironically France was led by Napoleon’s nephew Napoleon III.


Mathematician Pedigree: 名师出高徒
1. Littlewood-> Peter Swinnerton-Dyer
2. GH Hardy -> Ramajunian / Hua Luogeng
3. Louis Richard -> Evariste Galois, Charles Hermite
4. Gauss -> Dedekind, Riemann
5. Dirichlet -> Dedekind, Riemann
6. Charles Hermite -> Lindermann
7. Hua Luogeng-> Chen Jin-run
8. SS Chern -> Yau ST
9. Camile Jordan -> Felix Klein, Sophus Lie

The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions

Gauss saved by French Lady

Gauss and Sophie Germaine

Gauss was from Brunswick, now Germany. His king and academic sponsor was killed by Napoleon Army. Gauss had a French pen-friend “Mr. Brun” who was the famous French lady mathematician Sophie Germaine in disguise. She asked Napolean’s general not to kill Gauss. He spent remaining life in the University of Göttingen, which produced Klein, Hilbert, Riemann… Göttingen replaced Paris to be the World Center of Math until Hitler destroyed it.