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This abstract suggests that the bound may be lower:

http://jointmathematicsmeetings.org/amsmtgs/2160_abstracts/1...

From http://jointmathematicsmeetings.org/meetings/national/jmm201...

No. That abstract says that the bound it 12 if the Elliot-Halberstam conjecture holds. Under that assumption, the bound has been 16 since 2000. The trick that brings the number down from 16 to 12 under that assumption is the same that brings the proven number down to 600. This is actually described in the linked wired article.

Ah. Thanks for the clarification.

Holy cow. You could almost say (naively I suppose), "They're almost there!" Of course, who knows if the distance between 600 and 12 is infinitely easier than the distance between 12 and 11. Still ... Progress!

Also from the article: "Conceivably, Maynard said, someone with a clever sieve idea could push this limit as low as 6. But it’s unlikely, he said, that anyone could use these ideas to get all the way down to a prime gap of 2 to prove the twin primes conjecture." So seems like something completely different is needed to actually prove the twin primes conjecture.

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