# 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: