A report on Twin prime, Prime number and Yitang Zhang

Composite numbers can be arranged into rectangles but prime numbers cannot
Zhang in 2014
The Rhind Mathematical Papyrus
Plot of the absolute values of the zeta function, showing some of its features
The Gaussian primes with norm less than 500
The small gear in this piece of farm equipment has 13 teeth, a prime number, and the middle gear has 21, relatively prime to 13
The sieve of Eratosthenes starts with all numbers unmarked (gray). It repeatedly finds the first unmarked number, marks it as prime (dark colors) and marks its square and all later multiples as composite (lighter colors). After marking the multiples of 2 (red), 3 (green), 5 (blue), and 7 (yellow), all primes up to the square root of the table size have been processed, and all remaining unmarked numbers (11, 13, etc.) are marked as primes (magenta).
The connected sum of two prime knots
Construction of a regular pentagon using straightedge and compass. This is only possible because 5 is a Fermat prime.

A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43).

- Twin prime

The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved.

- Twin prime

These include Goldbach's conjecture, that every even integer greater than 2 can be expressed as the sum of two primes, and the twin prime conjecture, that there are infinitely many pairs of primes having just one even number between them.

- Prime number

On April 17, 2013, Zhang announced a proof that states there are infinitely many pairs of prime numbers that differ by 70 million or less.

- Yitang Zhang

This result implies the existence of an infinitely repeatable prime 2-tuple, thus establishing a theorem akin to the twin prime conjecture.

- Yitang Zhang

The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.

- Prime number
Composite numbers can be arranged into rectangles but prime numbers cannot

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