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.

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), (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}

image

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

image

Russian Math VZMSh

Israel Gelfand, the student of Kolmogorov (the Russian equivalent of
Gauss), created in 1964 the famous VZMSh, a national Math Correspondence School.

He wrote: “4 important traits which are common to Math, Music, and
other arts and sciences:
1st Beauty
2nd Simplicity
3rd Precision
4th Crazy ideas.

The Russian mathematicians also built special Math-Physics schools:
Moscow School #7, #2, #57 (one of the best high school in the world, http://www.sch57.msk.ru) Leningrad Schools #30, #38, #239 (Perelman studied here)

IMO Super-coach: Rukshin

Rukshin at 15 was a troubled russian kid with drink and violence, then a miracle happened: He fell in love with Math and turned all his creative, aggressive, and competitive energies toward it.

He tried to compete in Math olympiads, but outmatched by peers. Still he believed he knew how to win; he just could not do it himself.

He formed a team of schoolchildren a year younger than he and trained them.
At 19 he became an IMO coach who produced Perelman (Gold IMO & Fields/Clay Poincare Conjecture). In the decades since, his students took 70 IMO, include > 40 Golds.

Rukshin’s thoughts on IMO:

1. IMO is more like a sport. It has its coaches, clubs, practice sessions, competitions.

2. Natural ability is necessary but NOT sufficient for success: The talented kid needs to have the right coach, the right team, the right kind of family support, and, most important, the WILL to win.

3. At the beginning, it is nearly impossible to tell the difference between future (Math) stars and those who will be good (at IMO) but never great (Mathematician).

Perelman rejected Fields Medal & $1m Clay Prize

Grigory Perelman

1. Perelman first published his Poincaré Conjecture proof at this site: http://arxiv.org/

2. After the USA study trip where he was attracted by Prof Hamilton’s attempted proof with ‘Ricci Flow’, Perelman rejected a Harvard Professor job offer, returned to Russia to prove the Conjecture in isolation for 8 yrs as a low-pay free researcher.

Absorbing the problem in its entirety and then boiling it down to an essence that proved simpler than everyone had assumed.

3. In 2006 he rejected both Fields medal and US$1 m Clay Prize!

4. He won 1982 IMO Gold at 16 with full mark (42/42).
Perelman explained to the Math Olympiad jury his solution, who gave him full mark. Before the jury walked away, Perelman said: “Wait, I have 3 more solutions to this question!”

A sphere is simply connected because every loo...