Yitang Zhang's paper proves that there is a gap less than or equal to 70 million that occurs infinitely often. More precisely, the paper proves that if you take 3.5 million integers that do not form a complete residue system modulo any integer greater than 1, then one of the pairwise distances among these integers occurs infinitely often as the difference of two distinct primes. We do not know a single value, written down as a concrete number like 2014, that provably occurs as a difference of two distinct primes infinitely often.

P.S. Due to the efforts of Maynard, Pintz, Tao and other members of the PolyMath8 group, the 70 million above has been improved to 252, and the 3.5 million above has been improved to 51.

Yitang Zhang's paper proves that there is a gap less than or equal to 70 million that occurs infinitely often. More precisely, the paper proves that if you take 3.5 million integers that do not form a complete residue system modulo any integer greater than 1, then one of the pairwise distances among these integers occurs infinitely often as the difference of two distinct primes. We do not know a single value, written down as a concrete number like 2014, that provably occurs as a difference of two distinct primes infinitely often.

P.S. Due to the efforts of Maynard, Pintz, Tao and other members of the PolyMath8 group, the 70 million above has been improved to 252, and the 3.5 million above has been improved to 51.

Added. The current record is now 50 for the size of the tuple in the method, yielding the bound 246 for a gap size that occurs infinitely often among the primes. The relevant papers by the Polymath8 group have appeared:

New equidistribution estimates of Zhang type, Algebra & Number Theory 8 (2014), 2067-2199.

Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), no. 12, 83 pp.

Yitang Zhang's paper proves that there is a gap less than or equal to 70 million that occurs infinitely often. More precisely, the paper proves that if you take 3.5 million integers that do not form a complete residue system modulo any integer greater than 1, one of the pairwise distances among these integers occurs infinitely often as the difference of two distinct primes. We do not know a single value, written down as a concrete number like 2014, that provably occurs as a difference of two distinct primes infinitely often.

P.S. Due to the efforts of Maynard, Pintz, Tao and other members of the PolyMath8 group, the 70 million above has been improved to 252, and the 3.5 million above has been improved to 51.

Yitang Zhang's paper proves that there is a gap less than or equal to 70 million that occurs infinitely often. More precisely, the paper proves that if you take 3.5 million integers that do not form a complete residue system modulo any integer greater than 1, then one of the pairwise distances among these integers occurs infinitely often as the difference of two distinct primes. We do not know a single value, written down as a concrete number like 2014, that provably occurs as a difference of two distinct primes infinitely often.

P.S. Due to the efforts of Maynard, Pintz, Tao and other members of the PolyMath8 group, the 70 million above has been improved to 252, and the 3.5 million above has been improved to 51.