viXra Math Papers Publishing Site for Anybody

“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 k \geq 1

\boxed {(p - 1)^{p^k} \equiv -1 \mod {p^k}}

Special case: if p=2, k=1

General case :
p = p_1.p_2... p_j... for all pj satisfying the theorem.

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), (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.
1 +  \frac {3}{4}\sqrt {2}, \frac {1}{2} - 5 \sqrt {3}, ..., a+b\sqrt {n}


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



Field: Galois, Dedekind


Dedelind was the 1st person in the world to define Field:
“Any system of infinitely many real or complex numbers, which in itself is so ‘closed’ and complete, that +, – , *, / of any 2 numbers always produces a number of the same system.”

Heinrich Weber (1842-1913) gave the abstract definition of Field.

Field Characteristic

1. Field classification by Ernst Steinitz @ 1910
2. Given a Field, we start with the element that acts as 0, and repeatedly add the element that acts as 1.
3. If after p additions, we obtain 0 again, p must be prime number, and we say that the Field has characteristic p;
4. If we never get back to 0, the Field has characteristic 0. (e.g. Complex Field)

Example: GF(2) = {0,1|+} ; prime p = 2
1st + (start with 0):
0 + 1 = 1
2nd (=p) +:
1 + 1 = 0 => back to 0 again!
=> GF(2) characteristic p= 2

Galois Field GF(p)

1. For each prime p, there are infinitely many finite fields of characteristic p, known as Galois fields GF(p).

2. For each positive power of prime p, there is exactly one field.
(This is the only IMPORTANT Theorem need to know in Field Theory)
E.g. GF(2) = {0,1}

Math Game: Chinese 9-Ring Puzzle  (九连环 Jiu Lian Huan)

To solve Chinese ancient 9-Ring Puzzle (九连环) needs a “Vector Space V(9,K) over Field K”

finite Field K = Galois Field GF(2) = {0,1|+,*}
and 9-dimension Vector Space V(9,K):
V(0)=(0,0,0,0,0,0,0,0,0) ->
V(j) =(0,0,… 0,1,..0,0) ->
V(9)= (0,0,0,0,0,0,0,0,1)

From start V(0) to ending V(9) = 511 steps.