From the wired article:
The Sieve of Eratosthenes works perfectly to identify primes, but it is too cumbersome and inefficient to be used to answer theoretical questions. Over the past century, number theorists have developed a collection of methods that provide useful approximate answers to such questions.
“The Sieve of Eratosthenes does too good a job,” Goldston said. “Modern sieve methods give up on trying to sieve perfectly.”
GPY developed a sieve that filters out lists of numbers that are plausible candidates for having prime pairs in them. To get from there to actual prime pairs, the researchers combined their sieving tool with a function whose effectiveness is based on a parameter called the level of distribution that measures how quickly the prime numbers start to display certain regularities.
The level of distribution is known to be at least ½. This is exactly the right value to prove the GPY result, but it falls just short of proving that there are always pairs of primes with a bounded gap.
Since we don't try to prove that there is an infinite number of primechains, it should work, right?
