Twin primes conjecture
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Arenstorf published a purported proof of this conjecture, but a fatal defect was discovered in the proof and the paper was retracted.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.