# My favorite Fermat Little Theorem with Pascal Triangle

Fermat Little Theorem: For any prime integer p, any integer m When m = 2, Note: 九章算数 Fermat Little Theorem (m=2) Pascal Triangle (1653 AD France ）= (杨辉三角 1238 AD – 1298 AD) 1 4 6 4 1 => sum = 16= 2^4 (4 is non-prime) [PODCAST] https://kpknudson.com/my-favorite-theorem/2017/9/13/episode-4-jordan-ellenberg

# The Ring Z/nZ, Fermat Little Theorem, Chinese Theorem (French)

Revision: Modular Arithmetics (1/2) Fermat Little Theorem (1/2) Chinese Theorem (Note: This is the “RING” foundation of “The Chinese Remainder Theorem” which deals with remainders )

# 九章算术 & Fermat Little Theorem

500 BCE Confucius Period 孔子时代，Chinese ancient Math text “Nine Chapters of Arithmetic” 中国 (九章算术) already recorded the following: Fermat Little Theorem (17th century): Note: Computer Cryptography is based on two math theorems: Chinese Remainder Theorem and Fermat Little Theorem.

# Elegant Proof: Fermat Little Theorem

Let’s construct a necklace with p (p being prime) beads chosen from m distinct colors. There will be m^p permutations of necklaces less: 1. m permutations of necklaces with beads of same color; 2. Joining the 2 ends of the necklace to form a loop. For prime p beads, there will be p cyclic permutations of beads which are the same. Total distinct necklaces = (m^p […]

# Group: Fermat Little Theorem

Use Group to prove Fermat Little Theorem: For any prime p, Let Group (Zp,*mod p) = {1,2,3….p-1};    [*mod p= multiply modulo p] For any non-zero m in Zp, Since Zp isomorphic~ to the ring of co-sets  of the form m+pZ   (eg. Z2 ~ {0+2Z, 1+2Z} For any m in Z not in the […]

# Fermat Little Theorem

Fermat ℓittle Theorem (FℓT) ∀ m ∈ N, 1) p prime => = m mod (p) Note: Converse False [Memorize Trick: military police = military in the mode of police] Note: p prime, ∀m, if p | m, => m ≡ 0 mod (p) …(1) => multiply p times: ≡ 0 mod (p) …(2) Substitute (1) […]

# Fermat’s Little Theorem Co-prime Condition

It is confusing for students regarding the two forms of the Fermat’s Little Theorem, which is the generalization of the ancient Chinese Remainder Theorem (中国剩馀定理) — the only theorem used in modern Computer Cryptography . General: For any number a We get, If (a, p) co-prime, or g.c.d.(a, p)=1, then p cannot divide a, thus […]

# Euler’s and Fermat’s last theorems, the Simpsons and CDC6600

I am a fan of Fermat, not only because my university Alma Mater was in his hometown Toulouse (France) named after him “Lycée Pierre de Fermat (Classe Préparatoire Aux Grandes Ecoles) ” , but also the “Fermat’s Last Theorem” (FLT) has fascinated for 350 years all great Mathematicians including Euler, Gauss,… until 1993 finally proved […]

# 困扰了人类358年 费马大定理 Fermat’s Last Theorem

Keywords: Fermat’s Last Theorem (FLT): Pierre de Fermat (France 1637 AD): FLT Conjecture or Prank ? Euler (n= 3) Taniyama(谷山)-Shimura(志村)-(André) Weil Conjecture (now Theorem) Modular Form = Elliptic Curve Galois Group Symmetry Andrew Wiles (UK Cambridge 1994): (Modular Form = Elliptic Curve) <=> FLT (q.e.d.) Notes: 1. Do not confuse Prof Andrew Wiles (proved FLT) […]

# Our Daily Story #1 : The Fermat’s Last Theorem

While reading “Our Daily Bread” during my daily Bible reading time, it strikes me an idea to create a series of “Our Daily Story” for our Math studying time. The former makes the Bible alive, connected to our daily life in the context of scriptures; the later will make Math alive, motivate the interest and […]

# 少年学习＂去＂抽象化数学 Concretized De-Abstract Math

＂去＂抽象化 ＝Concretized De-Abstract Math 序言： 抽象数学 (Abstract Math) 是现代科学的工具， 通常是大学 理工 &非数学系 本科生undergraduate (大一/大二) 的必修数学。 然而， 高中之前的数学是从古希腊 (Ancient Greek) 到 牛顿时代（17 – 18世纪）的传统数学 (几何, 代数, 三角，牛顿微积分 ) 。 这是世界上第一本给中国和海外华人少年（介于 16 – 18岁, 初/高中）学习 19世纪后＂去抽象化＂的近代数学 (Modern Math) 。希望能够 帮助 高中生 “搭桥”(bridge) 顺利过渡学习大学数学的困难。 本书不采用数学教科课本的＂定律-推论-证明＂(Theorem-Lemma-Proof) 的形式， 而是 先从＂动机＂(Motivation)出发 ，讲诉数学家发现定理的历史背景，并举出大自然界中的具体(concrete) 例子，从而引导学生的兴趣和领悟。 读者必备知识 Prerequisites : 初中数学 (O-Level Math) 中文数学名词 基础英文 […]

# The most addictive theorem in Applied mathematics

What is your favorite theorem ? I have 2 theorems which trigger my love of Math : Chinese Remainder Theorem: 韩信点兵, named after a 200 BCE Han dynasty genius general Han Xin （韩信） who applied this modern “Modular Arithmetic” in battle fields. Fermat’s Last Theorem：The Math “prank” initiated by the 17CE amateur Mathematician Pierre de […]

# BM Category Theory II 1.1: Declarative vs Imperative Approach

Excellent lecture using Physics and IT to illustrate the 2 totally different approaches in Programming: Imperative (or Procedural) – micro-steps or Local 微观世界 [eg. C / C++, Java, Python…] Declarative (or Functional) – Macro-view or Global 大千世界 [eg. Lisp / Clojure, Scala, F#, Haskell…] In Math:  Analysis (Calculus)  Algebra (Structures: Group, Ring, Field, Vector Space, […]

# Why call Integer Z a Ring ?

​Does below picture looks like a ring, or a clock if it is Z12 = {1 2 3 … 12 = 0}. Integers (Z) have 3 operations : {+ – x} but not {÷} (or multiplicative inverse) – otherwise 2 integers divide would give a fraction (Q) which falls out of Z family.   An […]

# Russian & World Math Education

​In the world of Math education there are 3 big schools (门派) — in which the author had the good fortune to study under 3 different Math pedagogies: “武当派” French (German) ,  “少林派” Russian  (China) ,  “华山派” UK (USA). ( ) : derivative of its parent school. eg. China derived from Russian school in 1960s […]

# Homology: Why Boundary of Boundary = 0 ?

Homology and co-homology are the Top 10 Toughest Math in the world (1st & 8th topics in the list, of which 3rd, 4th, 9th and 10th topics received the Fields Medals). Like most Math concepts, which were discovered few decades or centuries ago, now become useful in scientific / industrial / computer applications never thought before by […]

# A General Proof of “The Theorem Wu”

“The Theorem Wu” as submitted by Mr. William Wu on the public Math Research Papers site viXra (dated 19-Nov-2014) can be further generalized as follows : The Theorem Wu (General Case) Prove that: if p is prime and p> 2 , for any integer [Special case: For p=2, k = 1 (only)] General case : […]