Smallest 5 Consecutive Primes in Arithmetic Sequence

Theorem
The smallest $5$ consecutive primes in arithmetic sequence are:
 * $9 \, 843 \, 019 + 30 n$

for $n = 0, 1, 2, 3, 4$.

Note that while there are many longer arithmetic sequences of far smaller primes, those primes are not consecutive.

Proof
But note that $9 \, 843 \, 019 + 5 \times 30 = 9 \, 843 \, 169 = 7^2 \times 200 \, 881$ and so is not prime.

Inspection of tables of primes (or a computer search) will reveal that this is the smallest such sequence.