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Two prime numbers separated by 2. For instance, 17 and 19 are twin primes. It is an open question if infinitely many twin primes exist.

Note that Euclid states a proof that infinitely many primes exist. It is also open to determine if there are infinitely many pairs of primes separated by other even numbers.

Mathematicians Dan Goldston and Cem Yildirim announced an important new result in late March 2003, which while not proving that there are infinitely many twin primes does substantially advance the understanding of "Small Gaps Between Primes", as their paper is called. It is said* to be one of the most important advances in prime theory for many decades.

On his homepage Goldston describes his current research as follows:

I have been working since 1985 on methods for proving that there are arbitrarily large primes that are unusually close together. The goal is to prove that for p and p' primes the infimum of the ratio (p-p')/ log p is zero. Here log p is the average distance between primes around p, and therefore we are trying to find consecutive primes within any fixed proportion of the average spacing. The current best result of Maier from 1986 shows that this ratio is infinitely often less than 0.248. Cem Yildirim and I are currently writting a series of papers on Higher Correlations of Short Divisor Sums which we hope will provide new tools both for small gaps between primes and other problems involving primes.

Early work by Hardy and Littlewood in the 1920s proved that if the General Riemann Hypothesis is true, then p' - p is infinitely often less than (2/3) log p. Later authors removed the dependence on unproved hypotheses and reduced the constant 2/3 to 0.248. What Goldston and Yildirim have now shown is that that for any ε > 0, there are infinitely many p with p' - p < ε log p.

Goldston's page: www.math.sjsu.edu/~goldston/publications.htm
Technical description: http://aimath.org/goldston_tech
Almost contentless press release of same: http://www.aimath.org/release_goldston.html
Earlier studies: http://www.cst.cmich.edu/units/mth/weekevents.htm

* though not by either of our experts ariels and Noether, who are both a bit dubious about the "importance".

The set of conjectures regarding cardinality of two-element prime constellations can be reduced, for each specific constellation, to a question of whether an example constellation can be found that is "large enough." A proof of this was proposed this past June, and is known to have passed some rounds of peer review; the current paper resides at https://drive.google.com/file/d/0B5KQFxR7gj3JdEFTTjEycE5JQVE/view.

The primary argument of this paper is that a sieve can be constructed which isolates the twin primes from the rest of the natural numbers, and this sieve can further adhere to a property described as the Leapfrog Lemma. In short, the purpose of this lemma is to encapsulate the Odd-Even Theorem and other properties of sieves in such a way so that an induction or other proof of infinity can be established, and then any sieve which has all the properties matching those found in this lemma is known to produce an infinite set of whatever object is under scrutiny.

While the Leapfrog Lemma made its first formal appearance only as recently as September 12th, 2017, the concept was certainly around almost as long as the Odd-Even Theorem, which in turn was discovered as early as February, 2001, and posted here the following October.

There is some speculation on a method for transforming the work in the above paper in order to apply the concepts of the main lemma to a wider selection of sieves. One of the current contenders is to change the function definitions so that the functions accept a vector input and return a set of values as the output. This particular approach could allow non-single-function sieves to be evaluated using the lemma, but brings about the question of whether sieves can be combined and retain valued properties, and if so, what limitations on this process might exist.

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