**“arXiv”** opposite is **“viXra”.**

The former **“arXiv”** is administered by Cornell University for Math paper publishing online. The traditional math journals would take 2 years to review and publish.

The Russian Mathematician **G. Perelman** was fed up of the long and bureaucratic review process, sent his proof of the 100-year-old unsolved “**Poincaré Conjecture”** to **arXiv** site. Later it was recognized to be correct, but Perelman refused to accept the **Fields Medal** and $1 million **Clay Prize**.

The new site **“viXra”** is open to anybody in the world, while “arXiv” is still restricted to academia.

This young Singaporean published his new found Math Theorem on **“viXra”** site:

**Prove that**: if **p** is prime and **p> 2** , for any integer

**Special case:** if p=2, k=1

**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

[By using the Binomial Theorem and Legendre’s Theorem.]

** My Alternate Proof [Hint] : ** by using Graduate Advanced Algebra “

**Galois Finite Field Theory**“:

Let q = p^k, where p prime and k >=1, it can be proved that GF(p^k) is the Field extension of GF(p).

**See the complete general proof** here:

**Note:** We say that p is the characteristic, k the dimension, of the Galois Field GF(p^k) of order (size) p^k.

**Example**: p = 3, k=2, 3^2=9

2^9 = 512 = -1 (mod 9)

** Definition**: (Without much frightening jargons, for a layman to understand): A

**Field**is a number structure which allows

**{+ , *}**and the respective opposite operations

**{-, ÷ }**.

*More intuitively, any Field numbers can be computed on a calculator with {+, -, ×, ÷} 4 basic operations. *

It is a German term ** Körper **, translated as Field (English), **Corps** (French), **域** (Chinese / Japanese).

**Examples of some standard Fields **: Rational numbers (**Q**), Real numbers (**R**), Complex numbers (**C**).

**Note 1**: This diagram below explains what it means by **Extension** (or Splitting) Field:

**Q** is Rational Number Field (**a**, **b** in **Q**)

By extension (or splitting) we obtain new sub-Fields : eg.

**Note 2:** **Characteristic** of GF(2), the Binary Field {0,1} is **2** because:

1+1 = 0 (1 add 2 times)

or **2** x (1) = 0

P.S. The ancient Chinese ‘magic’ game Chinese 9-Linked-Rings (九连环) is using the advanced Math Galois Field GF(2).

Baguenaudier Chinese Rings:

**K ** = Field = GF(2)

**p ** = 2 = characteristic of **K**

** k ** = 9 = dimension of

**K-vector space**