## Green-Tao Theorem

In 2004 Ben Green and Terence Tao proved that for any positive integer $$k$$, there exist arithmetic progressions of primes with $$k$$ terms:

Ben Green and Terence Tao, The primes contain arbitrarily long arithmetic progressions, Annales of Mathematics, 167 (2008), pp.481-547, doi: 10.4007%2Fannals.2008.167.481.

Note that the following facts.
• For any odd prime $$p$$, $$\{2,p\}$$ is an arithmetic progression of primes with length $$2$$. The common difference $$p-2$$ is odd and $$p+(p-2)=2(p-1)$$ is never a prime. Thus there is no arithmetic progression of primes whose length is more than $$2$$.
• For $$k=3,4,5,\dotsc$$ all the arithmetic progressions of primes with $$k$$ terms are sequences of odd primes, and their common differences are even. The multiples of even integers are \begin{align*} 2\times1=4\times3=6\times2=8\times4 & = 2, \\ 2\times2=4\times1=6\times4=8\times3 & = 4, \\ 2\times3=4\times4=6\times1=8\times2 & = 6, \\ 2\times4=4\times2=6\times3=8\times1 & = 8, \\ 2\times5=4\times5=6\times5=8\times5 & = 0 \end{align*} modulo $$10$$.
• The length of any arithmetic progression of primes starting with $$5$$ is at most $$5$$.
• If the common difference of an arithmetic progression of primes starting with $$1,3,7,9$$ is not a multiple of $$10$$, then its length is at most $$5$$.
We find some examples of arithmetic progressions of primes with lengh $$6$$ or more by using MATLAB.

 Length 6-10 green-tao.xlsx Length 6 gt06.m   gt06.csv Length 7 gt07.m   gt07.csv Length 8 gt08.m   gt08.csv Length 9 gt09.m   gt09.csv Length 10 gt10.m   gt10.csv The output of gt10.m on the display