2. Let $d$ be the common difference, then $p_{1}, p_{1}+d, p_{1}+2 d, p_{1}+3 d, p_{1}+4 d$ are all primes. If $2 \nmid d$, i.e., $d$ is odd, then one of $p_{1}+d$ and $p_{1}+2 d$ is even, and it is not a prime. If $3 \nmid d$, then one of $p_{1}+d, p_{1}+2 d, p_{1}+3 d$ is a multiple of 3 (they form a complete residue system modulo 3), which is a contradiction.

If $5 \nmid d$, then one of $p_{1}, p_{1}+d, \cdots, p_{1}+4 d$ is a multiple of 5, which can only be $p_{1}=$ 5, in this case the common difference $d$ is a multiple of 6.

And $5,11,17,23,29$ is a sequence of 5 primes in arithmetic progression, so, $p_{5}$ is at least 29.