Patterns in the Primes

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.

要查看或添加评论,请登录

Dr. Johar M. Ashfaque的更多文章

  • The Big 3 of Machine Learning Tasks

    The Big 3 of Machine Learning Tasks

    The "Big 3" machine learning tasks, which are by far the most common ones. They are: Regression Classification…

  • The Glashow-Salam-Weinberg Model

    The Glashow-Salam-Weinberg Model

    The spontaneous breakdown of symmetry in this renormalizable field theory results in massive Proca bosons that act as…

  • Knot Theory: Origins

    Knot Theory: Origins

    In 1867, after watching Tait perform experiments with smoke rings made of poisonous gases, Thomson concluded that the…

    3 条评论
  • Condensed Matter Theory: An Overview

    Condensed Matter Theory: An Overview

    Condensed matter physics is a branch of physics that investigates the physical phenomena associated with the many-body…

    1 条评论
  • Primality Testing: Pseudoprimes

    Primality Testing: Pseudoprimes

    Given an integer n, it is easy enough to construct a test that will certify the primality of n. If n is not prime then…

  • The Muon Anomalous Magnetic Dipole Moment

    The Muon Anomalous Magnetic Dipole Moment

    The magnetic dipole moment of the muon is a measure of the strength of its interaction with a magnetic field. The…

  • Game Theory

    Game Theory

    Game theory is a branch of applied mathematics and economics that studies strategic situations where there are several…

    1 条评论
  • Latin Squares & The Thirty-Six Officers Problem

    Latin Squares & The Thirty-Six Officers Problem

    Latin squares have a long and rich history, reaching back as far as the 12th century when Ahmad ibn Ali ibn Yusuf…

  • Gravitational Waves

    Gravitational Waves

    Gravitational waves are ripples in the fabric of space-time caused by some of the most energetic processes in the…

  • Neutrinos

    Neutrinos

    The three flavours of neutrinos we know about only feel the weak force and to such a limited degree that they barely…

社区洞察

其他会员也浏览了