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 the Julia Language.