70-million Bounded Gap Between Primes

Since Ancient Greek :

  1. Euclid had proved there are infinite primes.
  2. Sieve of Eratosthenes to enumerate the primes.
  3. Recent time 3 Mathematicians GPY attempted another Sieve method to find the bounded gap (N) of primes in infinity, but stuck at one critical step.
  4. Dr. YiTang Zhang 张益唐 (1955 -) spent 7 years in solitude after failure in academic career, in 2013 during a 10-min walk at the deer backyard of his friend’s house, he found an Eureka solution for the GPY’s critical step: \boxed { \epsilon = \frac {1} {168}} which gave the first historical bounded Gap (N) from an infinity large number to a limit of 70 million.

Notes:

  • Chinese love the number “8” \ba which sounds like the word prosperity 发 \fa (in Cantonese) . He could have instead used 160, so long as \epsilon is small.
  • The Ultimate Goal of the Bounded Gap (N) is 2 (Twin Primes Conjecture) .
  • The latest bounded gap (N) is reduced from 70-million to 246 from The PolyMath Project led by Terence Tao using Zhang’s method by adjusting the various values of \epsilon (analogous to choosing different sizes of the holes or ‘eyes’ of the Prime Sieve.)

A Graduate Level Talk by Dr. Zhang:

A Simpler Overview:

Mathematics: The Next Generation

Historical Backgroud:

Math evolves since antiquity, from Babylon, Egypt 5,000 years ago, through Greek, China, India 3,000 years ago, then the Arabs in the 10th century taught the Renaissance Europeans the Hindu-Arabic numerals and Algebra, Math progressed at a condensed rapid pace ever since: complex numbers to solve cubic equations in 16th century Italy, followed by the 17 CE French Cartersian Analytical Geometry, Fermat’s Number Theory,…, finally by the 19 CE to solve quintic equations of degree 5 and above, a new type of Abstract Math was created by a French genius 19-year-old Evariste Galois in “Group Theory”. The “Modern Math” was born since, it quickly develops into over 4,000 sub-branches of Math, but the origin of Math is still the same eternal truth.

Math Education Flaw: 本末倒置 Put the cart before the horse.

Math has been taught wrongly since young, either is boring, or scary, or mechanically (calculating).

This lecture by Queen Mary College (U. London) Prof Cameron is one of the rare Mathematician changing that pedagogy. Math is a “Universal Language of Truths” with unambiguous, logical syntax which transcends over eternity.

I like the brilliant idea of making the rigorous Math foundation compulsory for all S.T.E.M. (Science, Technology, Engineering, Math) undergraduate students. Prof S.S. Chern 陈省身 (Wolf Prize) after retirement in Nankai University (南开大学, 天津, China) also made basic “Abstract Algebra” course compulsory for all Chinese S.T.E.M. undergraduates in 2000s.

The foundations Prof Cameron teaches are centered around 4 Math Objects:

1. SET 集合
– Set is the founding block of the 20th century Modern Math, hitherto introduced into the world’s university textbooks by the French “Bourbaki” school (André Weil et al) after WW1.

Note: The last “Bourbaki” grand master Grothendieck proposed to replace Set by Category. That will be the next century Math for future Artificial Intelligence Era, aka “The 4th Human Revolution”.

2. FUNCTION 函数
– A vision first proposed by the German Gottingen School’s greatest Math Educator Felix Klein, who said Functions can be visualised in graphs, so it is the best tool to learn mathematical abstractness.

3. NUMBERS
– The German mathematician Leopold Kronecker, who once wrote that “God made the integers; all else is the work of man.”

– The universe is composed of numbers in “NZQRC” (ie Natural numbers, Integers, Rationals, Reals, Complex numbers). After C (Complex), no more further split of new numbers. Why?

4. Proofs
– reading and debugging proofs.

Example 1: Proof by Contradiction, aka Reductio ad Absurdum (Euclid’s Proof on Infinitely Many Prime Numbers)

image

Challenge the proof: Why ?
image
Induction intuitively by:
image

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Example 2: Proof by Logic
image

[Hint:]
By Reasoning (which is unconscious), most would get “2 & A” (wrong answer)

By Logic (using consciousness), then you can proof …
Correct Answer: 2 and B
Test on all 3 Truth cases below in Truth Table:
p = front side
q = back side

image

Recommended The Best Book on Abstract Algebra by Prof. Peter J. Cameron

Twin Prime Hero

http://simonsfoundation.org/features/science-news/unheralded-mathematician-bridges-the-prime-gap/

On April 17, 2013, a paper arrived in the inbox of Annals of Mathematics, one of the discipline’s preeminent journals. Written by a mathematician virtually unknown to the experts in his field — a 50-something lecturer at the University of New Hampshire named Yitang Zhang — the paper claimed to have taken a huge step forward in understanding one of mathematics’ oldest problems, the twin primes conjecture.

Editors of prominent mathematics journals are used to fielding grandiose claims from obscure authors, but this paper was different. Written with crystalline clarity and a total command of the topic’s current state of the art, it was evidently a serious piece of work, and the Annals editors decided to put it on the fast track.

Yitang Zhang (Photo: University of New Hampshire)

Just three weeks later — a blink of an eye compared to the usual pace of mathematics journals — Zhang received the referee report on his paper.

“The main results are of the first rank,” one of the referees wrote. The author had proved “a landmark theorem in the distribution of prime numbers.”

Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.

“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”

Mathematicians at Harvard University hastily arranged for Zhang to present his work to a packed audience there on May 13. As details of his work have emerged, it has become clear that Zhang achieved his result not via a radically new approach to the problem, but by applying existing methods with great perseverance.

“The big experts in the field had already tried to make this approach work,” Granville said. “He’s not a known expert, but he succeeded where all the experts had failed.”

20130522-202308.jpg

There are a lot of chances in your career, but the important thing is to keep thinking,” Zhang said.

http://nautil.us/issue/5/fame/the-twin-prime-hero?utm_source=feedly

Prime Secret: ζ(s)

Riemann intuitively found the Zeta Function ζ(s), but couldn’t prove it. Computer ‘tested’ it correct up to billion numbers.

\zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots

Or equivalently (see note *)

\frac {1}{\zeta(s)} =(1-\frac{1}{2^{s}})(1-\frac{1}{3^{s}})(1-\frac{1}{5^{s}})(1-\frac{1}{p^{s}})\dots

ζ(1) = Harmonic series (Pythagorean music notes) -> diverge to infinity
(See note #)

ζ(2) = Π²/6 [Euler]

ζ(3) = not Rational number.

1. The Riemann Hypothesis:
All non-trivial zeros of the zeta function have real part one-half.

ie ζ(s)= 0 where s= ½ + bi

Trivial zeroes are s= {- even Z}:
s(-2) = 0 =s(-4) =s(-6) =s(-8)…

You might ask why Re(s)=1/2 has to do with Prime number ?

There is another Prime Number Theorem (PNT) conjectured by Gauss and proved by Hadamard and Poussin:

π(Ν) ~ N / log N
ε = π(Ν) – N / log N
The error ε hides in the Riemann Zeta Function’s non-trivial zeroes, which all lie on the Critical line = 1/2 :

All non-trivial zeroes of ζ(s) are in Complex number between ]0,1[ along real line x=1/2

2. David Hilbert:

If I were to awaken after 500 yrs, my 1st question would be: Has Riemann been proven?’

It will be proven in future by a young man. ‘uncorrupted’ by today’s math.

Note (*):

\zeta(s)=1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots = \sum \frac {1}{n^{s}} …[1]

\frac {1}{2^{s}}\zeta(s) =  \frac{1}{2^{s}}(1+\frac{1}{2^{s}}+\frac{1}{3^{s}}+\frac{1}{4^{s}}+\dots)

\frac {1}{2^{s}}\zeta(s) =  \frac {1}{2^{s}}+ \frac{1}{4^{s}} + \frac{1}{6^{s}} + \frac{1}{8^s} +\dots … [2]

[1]-[2]:

(1- \frac{1}{2^{s}})\zeta(s) = 1+ \frac{1}{3^{s}} + \frac{1}{5^{s}} + \dots + \frac{1}{p^{s}} +\dots

\text {Repeat with} (1-\frac{1}{3^s}) \text { both sides:}

(1- \frac{1}{3^{s}})(1- \frac{1}{2^{s}})\zeta(s) = 1+ \frac{1}{5^{s}} + \frac{1}{7^{s}} + \dots + \frac{1}{p^{s}} +\dots

Finally,

(1- \frac{1}{p^{s}}) \dots (1- \frac{1}{5^{s}})(1- \frac{1}{3^{s}})(1- \frac{1}{2^{s}})\zeta(s) = 1

Or

\zeta(s) = \prod \frac {1}  {1- \frac{1}{p^{s}}}= \sum \frac {1}{n^{s}}

Note #:
\zeta(s) = \prod \frac {1}  {1- \frac{1}{p^{s}}}= \sum \frac {1}{n^{s}}

Let s=1
RHS: Harmonic series diverge to infinity
LHS:
\prod \frac {1}{1- \frac{1}{p}}= \prod \frac{p}{p-1}
Diverge to infinity => there are infinitely many primes p

English: Zero-free region for the Riemann_zeta...

English: Zero-free region for the Riemann_zeta_function (Photo credit: Wikipedia)

What is Ideal ?

Anything inside x outside still comes back inside

=> Zero x Anything = Zero

=> Even x Anything = Even

Mathematically,

1. nZ is an Ideal, represented by (n)

Eg. Even subring (2Z) x anything big Ring Z = 2Z = Even

2. (football) Field F is ‘sooo BIG’ that

(inside = outside)

=> Field has NO Ideal (except trivial 0 and F)

Why was Ideal invented ? because of ‘failure” of UNIQUE Primes Factorization” for this case (example):

6 = 2 x 3
but also
6=(1+\sqrt{-5})(1-\sqrt{-5})
=> two factorizations !
=> violates the Fundamental Law of Arithmetic which says UNIQUE Prime Factorization

Unique Prime factors exist called Ideal Primes: \mbox{gcd = 2} , \mbox{ 3}, (1+\sqrt{-5}), (1-\sqrt{-5})

Greatest Common Divisor (gcd or H.C.F.):
For n,m in Z
gcd (a,b)= ma+nb
Example: gcd(6,8) = (-1).6+(1).8=2
(m=-1, n=-1)

Dedekind’s Ideals (Ij):
6 =2×3= u.v =I1.I2.I3.I4 ;
u= (1+\sqrt{-5})
v=(1-\sqrt{-5})

Let gcd(2,u) = 2M+N.u
M,N in form of a+b\sqrt{-5}

1. Principal Ideals:
2M = (2) = multiple of 2

2. Ideals (nonPrincipal) = 2M+N.u

3. Ideal prime factors: 6=2 x 3=u.v
Let
I1= gcd(2, u)
I2=gcd(2, v)
I3=gcd(3, u)
I4=gcd(3, v)
Easy to verify (by definition):
I1.I2=(2)
I3.I4=(3)
I1.I3=(u)
I1.I4=(v)
=> Ij are prime & unique factors of 6=I1.I2.I3.I4
=> Fundamental Law of Arithmetic satisfied!

=>Ij “Ideal“-ly exist! hidden behind ‘compound’ (2,3,u,v) !

Verify : gcd(2, 1+√-5).gcd(2, 1-√-5)=(2) ?

Proof by definition:

[2m+n(1+√-5)][2m’+n'(1-√-5)]
=[2m+n+n√-5 ][2m’+n’-n’√-5]
= 4mm’+2mn’+2nm’+6nn’
= 2(2mm’+mn’+m’n+3nn’)
= 2M
= 2 multiples
= (2) = Principal Ideal

Field: Galois, Dedekind

Dedekind
(1831-1916)

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)

http://www.google.com.sg/imgres?imgurl=http://info.makepolo.com/uploadfile/2012/0723/20120723100653765.jpg&imgrefurl=http://info.makepolo.com/htmls/6/69/2669.html&h=400&w=533&sz=44&tbnid=ExodLfHv3cQjHM:&tbnh=91&tbnw=121&zoom=1&usg=__hsZaBecpPNdvTvguQbaQftCsXgo=&docid=qXMWtmo8A-vXEM&hl=en&sa=X&ei=f2NaUciqKYrOrQeT2YHgDw&sqi=2&ved=0CEsQ9QEwAg&dur=591

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.

Prime and Perfect Square

For all primes p ≠2, (a,b ∈Z)

p= a² + b² <=> p ≡ 1 mod 4


(2=1² + 1²)
5= 1² + 2² = 1 + 4 ≡ 1 mod 4
13= 2² + 3² = 4 + 9 ≡ 1 mod 4
17=1² + 4² = 1 + 16 ≡ 1 mod 4
29= 2² + 5² = 4 + 25 ≡ 1 mod 4
37= 1² + 6² = 1 +36 ≡ 1 mod 4

Notes:

1) Perfect squares (4, 9, 16, 25… ) ≡ 0 or ≡ 1 mod 4
2) Prime (4n+1) = a² + b²   (Euler took 7 yrs to prove)

3) Gauss expanded the proof to quadratic reciprocity (2 prime numbers p & q are linked by mod 4)

Prime Number

Prime Number

Prove we can always find an interval [p, q] wherein no prime exists, for any p, q ∈ N
Proof:
For any n ∈ N
Let H= (n+1)! = 1 x 2 x … x(n+1)
Let G = {H+2, H+3,. . . , H+(n+1) }
=> G composites (trivial)

Choose [p, q] such that  :
1. p the closest prime < H+2,
2. q the closest prime > H+(n+1)
=> no prime ∈ [p, q]

—n—————-n!-(H+1)-p[(H+2)———H+(n+1)]q——————

Fermat Little Theorem

Fermat ℓittle Theorem (FℓT)

∀ m ∈ N,
1) p prime => m^p= 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:
m^p≡ 0 mod (p) …(2)

Substitute (1) to (2): m ≡ 0
=> p prime, ∀ m
m^p≡ m mod (p)
=> No need (m, p) = 1 [co-prime]

E.g. (m, p) = (3, 2) =1
3^2= 9 ≡ 3 mod (2) ≡ 1 mod (2)
9 ≡ 1 mod (2)

E.g. (m, p) = (6, 2), p =2 |6
6^2= 36 ≡ 6 mod (2) ≡ 0 mod (2)
6 ≡ 0 mod (2)

WRONG EXAMPLE:  p = 4 = non prime
2^4= 16 ≡ 2 mod (4)
but 16 ≡ 0 mod (4)
Equivalent:
p prime => m^{p-1}= 1 mod (p)

by Contra-positive:
3) m^{p-1} ≠ 1 mod (p)
=> p non-prime

Apply: Prove 39 non-prime?
Take m =2, p= 39
Prove m^{p-1} ≠ 1 mod (p)?
a) 2^{38} =2^{36}.2^2 .
b) 2^6= 64 ≡ 25 mod (39)
c) 25^2 = 625 ≡ 1 mod (39)
From (b) & (c):

(2^6)^6 = 2^{36}25^6 mod(39)≡(25^2)^31^3 ≡ 1
From (a): 2^{38} = 2^{36}.2^2 ≡ 1×4 ≠ 1 mod(39)
=> 39 non-prime