# Is there a known bound in prime gaps?

Is there known to be an x such that for all positive integers N there exists some n>N such that p_{n+1}-p_n < x, where p_n is the nth prime? Or, in other words, is it known that limit as n goes infinity of p_{n+1}-p_n is not infinity? If such an x is known to exist, what is the current best known x? (showing x=2 would imply the Twin Prime conjecture, of course)

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The short answer is no, though if one assumes the Elliot-Halberstam conjecture then one can take x=16. See

http://arxiv.org/abs/math/0605696

for a comprehensive survey of the best known results (both conditional and unconditional).

There is also the Wikipedia article at

http://en.wikipedia.org/wiki/Prime_gap

although this is less comprehensive.

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Assuming this is not wrong, the short answer is now 70 million.

See the current front page of the AMS ...

Bounded Gaps Between Primes Wednesday May 15th 2013

That is the title of a paper by Yitang Zhang (University of New Hampshire) that shows that there are infinitely many pairs of consecutive prime numbers that differ by a finite distance. The paper, now posted on the Annals of Mathematics web site, represents progress on settling the Twin Prime Conjecture, which states that there are infinitely many prime numbers that differ by 2. Zhang's paper (available to subscribers) shows that there are an infinite number of consecutive primes that differ by less than 70 million. In an article in New Scientist, Henry Iwaniec (Rutgers University) said that the result is "beautiful." Nature also has an article about Zhang's result. Read Evelyn Lamb's post about the proof and about a proof of the Weak Goldbach Conjecture by Harald Helfgott (École Normale Supérieure) in "This Week in Number Theory" in the new AMS Blog on Math Blogs.

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