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 )

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 )

“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** :

for all pj satisfying the theorem.

**Examples:**

p = 9 = 3×3

p = 21= 3×7

p = 27 = 3×9 = 3x3x3

p =105 = 3x5x7

p =189 = 3x7x3x3

[Mr. William Wu proved the non-general case by using the Binomial Theorem and Legendre’s Theorem]

I envisage below to prove for **all cases**

by using the Advanced Algebra **“Galois Finite Field Theory”** :

Let

where p prime and k >=1, the **Fundamental Theorem of Galois Finite Field** states that

1. The Galois Finite Field GF(p) is a multiplicative cyclic group;

2. GF(q) is the Galois Field extension of GF(p).

**Step 1**: **Identity Equation**:

**Proof:**

Except the first term X^q and the last term Y^q (both with coefficient 1), all the middle terms with coefficients as :

are divisible by q,

thus,

[QED]

**Step 2:**

Apply the Identity Equation:

we get,

….. [*]

Since GF(q) is a multiplicative cyclic group of order q, we get

[QED]

**Step 3: General case**

Let

Apply the Step 2 equation [*],

[QED]

**Lemma :**

**Proof:** By The Fundamental Theorem of Arithmetic, any number (**even** and **odd** numbers) can be factorized as a product of primes.

Since The Theorem Wu is true for **primes** only,

=> it is false for **even** numbers

=> by the General case above, it is true for all **odd numbers**.

__ Corollary__ (

If m an **even** number,

Note: **Wolfram Alpha verification **

For p any prime or product of primes:

**Galois Fundamental Finite Field Theorem**

**Note:**

This is an excellent example of marriage of ancient Chinese Math, Middle-Age French Maths (Fermat & Galois) and modern Chinese (Singaporean) Maths — across 2,000 years!

**The Theorem Wu** is more general than the Ancient Chinese Remainder Theorem in 2 CE, which was further improved by 17 CE Fermat’s Little Theorem.

The Theorem Wu proves any number **is/isn’t**

1) a prime or

2) a product of primes > 2 (this property is neither found in Chinese Remainder Theorem nor Fermat’s).