It is widely believed that sieves can not detect prime numbers. We discuss how to adapt sieve methods in certain cases provided that some additional properties of the sequence considered are known — this particular adaptation is now known as Bombieri asymptotic sieve.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/7671