In the sequence of prime numbers, some adjacent primes have a difference of 2, such as 3 and 5, 5 and 7, 11 and 13, and so on. Such primes are called twin primes. At first glance, it seems that twin primes are quite dense in the sequence of prime numbers, but in reality, it's not the case. It becomes increasingly difficult to encounter them as we go further. It can be proven that twin primes are quite sparse in the sequence of prime numbers, with a density of zero; that is, almost all prime numbers are not twin primes. Despite this, twin primes still appear from time to time. In 2011, two twin primes with 200700 digits were discovered: 3756801695685 × 2666669 ± 1. It seems that twin primes will never end. It is conjectured that there are infinitely many pairs of twin primes. The difficulty of this conjecture is comparable to that of Goldbach's conjecture for even numbers, and it has not been solved yet. In 2013, Zhang Yitang, a Chinese mathematician living in the United States, proved that there exists a positive even number n, not exceeding 7×10^7, such that there are infinitely many pairs of prime numbers p and q satisfying p q = n. This is the most outstanding research progress since the twin prime conjecture was proposed.
The twin prime conjecture can also be stated as follows: the polynomials x and x + 2 can simultaneously take prime values infinitely many times. Using the language of polynomials, the triple prime conjecture can also be proposed: the polynomials x, x + 2, and x + 6 can simultaneously take prime values infinitely many times. More generally, British mathematicians Hardy and Littlewood proposed the k-prime conjecture, where k is a fixed positive integer. Of course, none of the above conjectures have been proven.
The Erdős–Turán conjecture considers some connections between prime numbers from another perspective. It does not have specific requirements for the difference between prime numbers (such as the difference of twin primes being 2), but requires an arbitrary number of prime numbers. The Erdős–Turán conjecture states that there exists an arbitrarily long arithmetic subsequence in the sequence of prime numbers. In 2004, Tao Zhuoxuan and Green combined ergodic theory and analytic number theory to solve this long-standing problem in number theory, which was also one of Tao's main contributions to winning the Fields Medal.