What are the shortest intervals between

consecutive prime numbers? The

twin primes conjecture, which asserts that P

_{n+1}-P

_{n}=2 (where P

_{n}, as usual, stands for the N-th

prime) infinitely often is one of the oldest problems; it is difficult to trace its origins.

In the 1960's and 1970's sieve methods developed to the point where the great Chinese mathematician Chen was able to prove that for infinitely many primes P the number P+2 is either prime or a product of two primes. However the "parity problem" in sieve theory prevents further progress.
What can actually be proven about small gaps between consecutive primes? A restatement of the prime number theorem is that the average size of P_{n+1}-P_{n} is log(P_{n}). A consequence is that Δ := lim_{n → ∞} P_{n+1}-P_{n} ≤ 1

In 1926, Hardy and Littlewood, using their ``circle method'' proved that the Generalized Riemann Hypothesis (that neither the Riemann zeta function nor any Dirichlet L-function (?) has a zero with real part larger than 1/2) implies that Δ ≤ 2/3. Rankin improved this (still assuming the Riemann Hypothesis) to Δ ≤ 3/5. In 1940 Erdös, using sieve methods, gave the first unconditional proof that Δ ≤ 1. In 1966 Bombieri and Davenport, using the newly developed theory of the large sieve (in the form of the Bombieri - Vinogradov theorem) in conjunction with the Hardy - Littlewood approach, proved unconditionally, Δ ≤ 1/2 and then using the Erdös method they obtained Δ ≤ (2 + "√ 3)/8 (or about 0.46650). In 1977, Huxley combined the Erdös method and the Hardy - Littlewood, Bombieri - Davenport method to obtain Δ ≤ 0.44254. Then, in 1986, Maier used his discovery that certain intervals contain a factor of more primes than average intervals. Working in these intervals and combining all of the above methods, he proved that Δ ≤ 0.2486, which was the best result until now.

However, Dan Goldston and Cem Yildirim have recently written a manuscript (which was presented in a lecture at the American Institute of Mathematics) which advances the theory of small gaps between primes by a huge amount. First of all, they show that Δ = 0, exactly. Moreover, they can prove that for infinitely many n the inequality P_{n+1}-P_{n} < {log(p_{n})}^{8/9}holds.

Goldston's and Yildirim's approach begins with the methods of Hardy-Littlewood and Bombieri - Davenport. They have discovered an extraordinary way to approximate, on average, sums over prime K-tuples. We believe, after work of Gallagher using the Hardy-Littlewood conjectures for the distribution of prime K-tuples, that the prime numbers in a short interval ( N, N+ Γlog(N) ) are distributed like a Poisson random variable with parameter Γ. Goldston and Yildirim exploit this model in choosing approximations. They ultimately use the theory of orthogonal polynomials to express the optimal approximation in terms of Laguerre polynomials. Hardy and Littlewood could have proven this theorem under the assumption of the Generalized Riemann Hypothesis; the Bombieri - Vinogradov theorem allows for the unconditional treatment.

This new approach opens the door for much further work. It is clear from the manuscript that the savings of an exponent of 1/9 in the power of log(P_{n}) is not the best that the method will allow. There are (at least) two possible refinements. One is in the examination of lower order terms that arise in his method. Can they be used to enhance the argument? The other is in the error term Gallagher found in summing the ``singular series'' arising from the Hardy-Littlewood K-tuple conjecture. There is reason to believe that this error term can be improved, possibly using ideas in recent work of Montgomery and Soundararajan (``Beyond Pair - Correlation''.)

This work is astonishing in another regard. They have actually proven that for any fixed number r the inequality
P_{n+r}-P_{n} < (Log(P_{n})^{8r/(8r+1)}) holds for infinitely many n.

In 1977 Helmut Maier burst onto the analytic number scene with his tour de force proof that the largest gaps known to hold for two consecutive primes could be proven for each gap of r consecutive gaps for any fixed r. Goldston and Yildirim have achieved a similar sort of result for consecutive small gaps at the same time, in that they have demolished all previous records for one small gap.

It is clear is that at least part of a monumental barrier which has impeded progress for at least the last 80 years has been broken down, and the paper will allow a significant amount of research to go forward.