Patterns in the Primes

It has long been known that there are infinitely many primes and that there are infinitely many primes in any arithmetic progression
a, a+d, a+2d,...
provided \gcd (a,d)=1 and d is greater than or equal to 1.

If we ask some in depth questions such as whether there exist infinitely many primes of the form n^{2}+1 or infinitely many primes of the form (p, p+2) then these are open ended questions.

Any two primes form an arithmetic progression of length two. Then the first non-trivial question one may ask is whether there exist many three-term arithmetic progressions of primes that is triples of the form
a, a+d, a+2d
where a and d are non-zero integers. In 1933, van der Corput proved that there are indeed infinitely many such triples of distinct primes and there have been several different proofs of this thereof. However, the question as to whether one can have four or more primes occurring in arithmetic progression infinitely often had seemed far beyond the reach of current methods of mathematical research until very recently.

The situation changed in 2005, beginning with the revolutionary paper of Green and Tao. Using a panorama of new ideas they showed that for any integer k there are infinitely many k-term arithmetic progressions of primes that is there exist infinitely many distinct pairs of non-zero integers a and d such that
a, a+d,..., a+(k-1)d
are all prime. The work is based on ideas from many fields like harmonic analysis, ergodic theory, additive number theory, discrete geometry, and the combinatorics of set theory. As a consequence a new mathematical discipline came into existence namely additive combinatorics and has been since used to prove new major results in graph theory, group theory and theoretical computer science and not to forget analytic number theory.

It is often the case that the conjectures made in mathematics lie just beyond the horizon of what has been well established. There have been many conjectures regarding the distribution of primes in various sequences. However, the horizon was greatly extended by the wonderful breakthrough of Green and Tao that little is known of what lies beyond it.

Where Green and Tao proved the existence of numerous patterns, Granville sought to find examples of each of these patterns. He showed how the results of Green and Tao generate all sorts of mathematical and aesthetically desirable patterns.

Definition: An arithmetic progression of primes is a set of primes of the form p_{1}+kd for fixed p_{1} and d and consecutive k that is
p_{1}, p_{1}+d,  p_{1}+2d,...
As an example the smallest arithmetic progression of length 5 is given by               5, 11, 17, 23, 29

Note: ``Smallest" refers to the largest prime in the arithmetic progression being the smallest.