FTA Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic | by Maths and Musings | Cantor’s Paradise



The Fundamental Theorem of Arithmetic (FTA):

2300 years ago Euclid 《Elements》proved prime factorisation :
If p | x. y then p | x or p |y

19CE Gauss proved FTA. Uniqueness of prime factorisation.

French 《Bezout Theorem》 is the useful tool: if p, q co-prime, then there exist n, m integers such that
np + mq = 1
eg. Prove 5, 7 co-prime,
(3) *5+ (-2) *7 = 15-14=1

Strange, why Bezout Theorem not taught in A level ?

新数启蒙 (1-5) – 给16岁以上的中学生

This “Modern Math” Introductory Course is based on the French Baccalaureate “Modern Math” for 16+ years old, simplified & customized for local students in Chinese.


(1)集合: 等价关系 Equivalence Relation


(2)集合与集合的关系:映射 Mapping


(3) 新数 历史: Bourbaki 学派


(4)由外向内看代数结构 : 域Field,环 Ring,群 Group,向量空间 Vector Space


(5) 域 Field : 足球场


Modern Math Education : Time for Reform

Modern Math Foundation : Set + Logic (Equivalence Relation). 

A level (H2 level) / International Baccalauréat Math should study the big red square box (Number System, Set Theory, Logic).

All STEM university Year 1 & 2 study these 3 Math foundations : Algebra, Analysis & Vector Space.

The Math major university Year 3&4 study more : real /complex analysis, Topology, Galois Theory, graph theory, etc…


mRNA剪接Splicing 原理 – 施一公

3 Step Splicing : first & last steps are linear, 2nd step 3D syructure (non-linear) .

施一公 returned from USA, established the private 西湖大学 (funded by Alibaba, Baidu, Huawei etc), a first pure research university modeled on Caltech. His team first decoded the COVID19 genetic code sequence just 1 month after the Wuhan outbreak & announced the result FREE to the world.

The Hungarian lady scientist Katalin Karikó jumped on his newly published DNA code to invent the mRNA PfizerBioNTech vaccines.




Math was a tool for Physics (Newtonian Physics)

Math & Physics were independently developed ( Physics Gauge Field = Math Fibre Bundle )

Physics is a tool for discovering Math (Quantum Physics : String Theory)

杨振宁 Yang-Mills Conjecture is one of the unsolved Millenium Math Problems.


Feymann Calculus Trick

理查德·费曼非常聪明的求导 differentiate 技巧

Feymann (Nobel Physicist) has many funny speedy Math tricks for Calculus eg. Differentiate an Integral (Applied Fundamental Theorem of Calculus) , and this one below.





中科院数学院士 林群:"如何学高深 抽象 数学":
先通过案例 (case / motivation) 学道理,后才学证明 (Theorem Proofs) 。

eg. Learn “Group Theory”
Case / Motivation: Symmetry


Ellipse Proof by Circle : Affine Transformation

Cumbersome Ellipse proof below :

By Affine Transformation the Ellipse to a Circle, the corresponding ratios preserved. Hence it is easier to prove in the circle below:

Similar triangles CQV ~ CTQ


圆幂 定理 II: The Affine Transformation FROM Circle to Ellipse. Also applicable to all other 2 Theorems.


Proof : Ellipse Area from Circle :

函数 Function 概念


Function = f : R – > R
(with 1& only 1 image)

Mapping 映射 (l’Application) , can be non-value to non-value

仿射变换 Affine Transformation (解析几何 Analytic Geometry 的 Modern Math).

Affine 是 德文 (coined by Euler)


丘成桐 的雄心计划: 培养800个数学天才苗子

当年汉朝 霍去病 带 800 骑士 深入漠北驱赶匈奴。

丘成桐 (属牛)的雄心计划: 培养800个数学天才苗子, 每年从中国&世界选16岁起的100个 华人,在清华少年班 8年培训:本科4年+硕2年+博士 2年.