Constructing arithmetic progressions It is known that in the sequence of primes there exists arithmetic progressions of primes of arbitrary length. This was proved by Ben Green and Terence Tao in 2006. However the proof given is a nonconstructive one.
I know the following theorem from Burton gives some criteria on how large the common difference must be.

Let $n > 2$. If all the terms of the arithmetic progression
  $$
p, p+d, \ldots, p+(n-1)d
$$
  are prime numbers then the common difference $d$ is divisible by every prime $q <n$.

So for instance if you want a sequence of primes in arithmetic progression of length $5$ ie
$$
p, p+d, \ldots, p+4d
$$
you need that $d \geq 6$. Using this we can get that the prime $p=5$ and $d = 6$ will result in a sequence primes in arithmetic progression of length $5$.
So my question is what are the known techniques for constructing a sequence of primes of length $k$? How would one find the "first" prime in the sequence or even the "largest prime" that would satisfy the sequence (assuming there is one)? Also, while the theorem gives a lower bound for $d$, is it known if it is the sharpest lowest bound there is?
NOTE: This is not my area of research so this question is mostly out of curiosity.

This may not answer the question, but I would like to point out that more recent work of Green and Tao have proven even stronger results.  
Specifically, Green and Tao give exact asymptotics for the number of solutions to systems of linear equations in the prime numbers, and their paper Linear Equations in the Primes was published in the Annals in 2010.  In particular, this tells us asymptotics for the number of $k$-term arithmetic progressions in the primes up to $N$.
For example, as $N\rightarrow \infty$, we can count the asymptotic number of 4-tuples of primes $p_1<p_2<p_3<p_4\leq N$ which lie in arithmetic progression, and it equals $$(1+o(1))\frac{N^2}{\log^4(N)} \frac{3}{4}\prod_{p\geq 5} \left(1-\frac{3p-1}{(p-1)^3}\right).$$
Do note however, that Green and Tao's paper made two major assumptions.  They assumed the Möbius Nilsquence conjecture (MN) and the Gowers Inverse norm conjecture (GI).  In a paper published in the Annals in 2012, Green and Tao resolve the MN conjecture, proving that the Möbius Function is strongly orthogonal to nilsequences.  Recently, Green, Tao and Ziegler resolved the Gowers inverse conjecture, and their paper is currently on the arxiv.  (It has not yet been published)  This means that we unconditionally have asymptotics for the number of primes in a $k$ term arithmetic progression.
If you would like to learn more, I suggest reading Julia Wolf's excellent survey article Arithmetic and polynomial progressions in the primes, d'après Gowers, Green, Tao and Ziegler.  It is very recent, as it was for the Bourbaki lectures at 2 month ago.
