Prime Number Theory This is a remarkable story of dedication.  Last summer, Yitang Zhang proved "a landmark theorem in the distribution of prime numbers", relating to the Twin Primes Conjecture.  “Basically, no one knows him, ... now, suddenly, he has proved one of the great results in the history of number theory.”

Full quote:
       “The main results are of the first rank,” one of the referees wrote. [But who was the author?]  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.”

What did Yitang Zhang prove? To understand this, one needs a bit of background, but we can get you there in 10 steps...

Definition. A prime number is any number that is divisible only by 1 and itself, i.e. has no other factors than these. E.g. 2,3,5,7,11,13, but not 9 (=3x3) or 14 (=2x7), etc.

1. Euclid (of Alexandria, Egypt, 325-265 BCE) proved the infinitude of the primes with a famous proof by contradiction.

2. The Twin Prime conjecture is the statement that there are infinitely many prime pairs with a gap of two, e.g. 3,5, or 11,13.

3. The Generalized Twin Prime conjecture is the statement that there are infinitely many prime pairs with a finite gap, but we don't know how large the gap is.

4. Eratosthenes (of Cyrene, Libya, 276-194 BCE) invented the Sieve of Eratosthenes to find prime numbers up to a given arbitrary number. This is the basic and most ancient of prime sieves (from 2000 years ago).
https://www.famousscientists.org/eratosthenes/

5. Lucas showed M127 (39-digit number) is prime, the Mersenne number of form 2^n -1, n=127, which is the largest known prime discovered without the use of electronic methods, a record which lasted till 1952 and Robinson's use of an early computer to show that M2281 is prime. By 2018, we know of 50 Mersenne primes, the largest of which is M77 232 917, which has just under 23 million 250 thousand digits. [1]

6. The largest known twin prime pair is 2 996 863 034 895 × 21 290 000 ± 1, with 388 342 decimal digits, discovered in September 2016. [1]

7. The level of distribution (LoD) is a parameter that measures how quickly the prime numbers start to display regularities, and is known to be at least 1/2.

8. In the 1986, Bombieri, Friedlander, and Iwaniec showed that they could bring the value of LoD to 4/7 by a slightly tweaked definition. [3]

9. In 2005, Goldston, Pintz, and Yildirim showed that there are prime pairs for any fraction of the average gap between primes, no matter how small the fraction. They did so using the GPY sieve that filters out pairs of primes that are closer together than the average, i.e. that are plausible candidates for having prime pairs in them, and LoD value of 1/2. They also showed that if LoD could be shown to be larger than 1/2, any amount larger, then this statement could be strengthened that there are always prime pairs with at most a bounded gap with a definite bound (bounded prime gap conjecture). [4]

This is where Zhang's result comes in...

10. In 2013, Yitang Zhang sent in a paper solving the Generalized Twin Prime conjecture (#3 above) for a gap of 70 million, i.e. given any prime, there will be another prime within at most the next 70 million numbers. Since there are infinitely many primes, this means that there are infinitely many prime pairs with gap closer than 70 million numbers. He achieved this using a modified GPY sieve, that filters using only numbers that don't have a large prime factor, i.e. using numbers with small prime factors.

Andrew Granville: "This work is a game changer, and sometimes after a new proof, what had previously appeared to be much harder turns out to be just a tiny extension. [Zhang] nailed down every detail so no one will doubt him. There's no waffling."


References

[1] History of Prime numbers:
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html#7

[2] References for Prime number history:
http://www-groups.dcs.st-and.ac.uk/history/HistTopics/References/Prime_numbers.html

[3] 1986: LoD tweak to 4/7:
https://link.springer.com/article/10.1007%2FBF02399204

[4] 2005: GPY
https://arxiv.org/abs/math/0508185