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:

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张益唐:我的数学人生

[录音小声, 请用earphone耳机听更清楚。]

Key Points Take Away:

1. 身处逆境, 不是勇气, 是淡定。

2. 对目的要穷追不捨, 不要放弃。他从北大的Analytic Number Theory (解析数论)兴趣, 被”人为”的转道去搞博士论文Algebraic Geometry, 7年毕业却无业。从新回到” 解析数论”的跑道, 才得到大成就。

3. 如果2个不同领域的学问之间有些联系, 只要往里鑽, 必能发现新东西。

4. 人生低谷, 碰到3个贵人(2位北大校友, 一位美国系主任青睐)协助。

5. 太太不知他干何学问, 不给 他家庭经济压力, 才能安心于数学。

Q&A:
1. 对于天才儿童, 他劝家长不要 “压 “也不要”捧”, 只要多鼓励, 像Perleman 的(俄国数学家, 证明100年的Poincaré Conjecture)父母循循教导儿子

2. 希望能收PhD学生, 会对他们负责任, 不要有像他个人的悲剧发生 (指被教授利用做私人的项目, 误了学生的前途)。他手头有半’成品’和 3/4’成品’, 可让学生拿去参考, 继续完成当论文。

张益唐: 速食店员竟然是数学天才

【台湾壹週刊】

速食店员竟然是数学天才

张益唐 (1955 – ) : 北京大学 – 美国数学博士。因为执着数学理论的真理, 得罪美国大学台湾籍论文教授, 毕业后找不到大学教职, 在朋友的 Subway 速食店做会计8年, 潜心业余思考世界数学大难题: Twin Primes Gap, 终于攻破。

他的下一个目标是Riemann Hypothesis, 困扰数学家百年的难题: “素数 (Prime numbers)的分布”都集中在 Zeta function complex plane的 实轴(real = 1/2) 上。大数学家David Hilbert说如果五百年后复活, 第一件事会急着问 “Riemann Hypothesis” 证明了吗?

张益唐谈做数学

2013年7月13日 台大访问笔记摘要 Summary:

http://blog.sina.cn/dpool/blog/s/blog_c24597bf0101ctdp.html

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突破瓶頸: “先上对车, 后补上票”

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Holistic Approach to Attack Math :

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新酒进旧瓶, 可以突破: 勤能补拙

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10岁的启蒙书:

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现代”科举”考场失意:

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文学与数学相通: Intuition

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Ref: 白居易写给元稹《与元九书》

如何教好数学?

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Shimura Modular Form:

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好书推荐: 华罗庚的《数论导引》 , 华的剑桥老师Hardy…

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解析数论 Analytic Number Theory:

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选对导师和有兴趣的题目
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On Riemann Hypothesis:

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Our Daily Story #6: A Subway Sandwich Mathematician Zhang Yitang 张益唐

Zhang is the typical demonstration of pure perseverance of traditional Chinese mathematicians: knock harder and harder until the truth is finally cracked.

His work is based on the prior half-way proof by 3 other mathematicians “GPY”:

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Gap between Primes:

Let p1 and p2 be two adjacent primes separated by gaps of 2N:

p1 – p2 = 2 (twin primes)
eg. (3, 5), (5, 7)… (11, 13) and the highest twin primes found so far (the pair below: +1 and -1)
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p1 – p2 = 4 (cousin primes)
eg. (7, 11)

p1 – p2 = 6 (sexy primes)
eg. (23, 29)

p1 – p2 = 2N

Euclid proved 2,500 years ago there are infinite many primes, but until today nobody knows are these primes bounded by a gap (2N) ?

Zhang, while working as a sandwich delivery man in a Subway shop, kept trying alone for 7 years with the GPY method, finally in 2013 he found the bounded gap: \boxed {2N <  70,000,000,000}

Recent mathematicians (Terrence Tao’s Polymath Project and others) follow his technique to narrow 2N down from 70 million to, hopefully, 16.

At age above 50, Zhang (1955 -) has shown that Mathematics is not limited to only young minds – as GH Hardy had set 35 the age limit for any great math achievements.

http://en.m.wikipedia.org/wiki/Yitang_Zhang

Part 1:

Part 2:

Ref:

1.
https://tomcircle.wordpress.com/2013/05/22/prime-gap-by-unknown-mathematician/

2. 张益唐(中文) 伯克利大学演讲记:
http://bluepanda5.wordpress.com/2014/02/17/%e5%bc%a0%e7%9b%8a%e5%94%90%e5%9c%a8%e4%bc%af%e5%85%8b%e5%88%a9%e7%9a%84%e6%bc%94%e8%ae%b2%e7%b5%ae%e8%ae%b0-%ef%bc%88%e8%bd%ac%e8%bd%bd%ef%bc%89/

3. Euclid’s Proof of Infinite Primes: