Twin primes conjecture
The twin primes conjecture states that there are infinitely many twin primes. It is one of the great unsolved problems in mathematics, as it has never been proven.
In 2004, Professor Arenstorf published a purported proof of this conjecture, but a fatal defect was discovered in the proof and the paper was retracted.
In 1966, Chen Jingrun showed that there are infinitely many primes p such that p+2 is either a prime or the product of two primes (a "semiprime"), relying on a sieve theory. This resulted in defining a "Chen prime" to be a prime p such that p+2 is either a prime or a semiprime. In 2005, Ben Green and Terence Tao proved that there are infinitely many three-term arithmetic progressions of Chen primes.
In 1915, Viggo Brun proved that the sum of reciprocals of the twin primes is convergent. This sum converges to Brun's constant.