Zhang proved a bound of 70,000,000. A flurry of activity has incrementally reduced this to a mere 4,788,240, an order of magnitude improvement -- only another 6 orders to go!
For that new limit, see comment 68 in Scott Morrison's post, linked to in part 1.1 of the Polymath proposal post. Reading through all the comments is highly educational about this kind of collaborative work -- the excitement buzzes through.
UPDATE 20/6/2013: Michael Nielsen has been keeping a record of the records: the value is now down to ~60,000
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