Smallest 18 Primes in Arithmetic Sequence

Theorem
The smallest $18$ primes in arithmetic progression are:
 * $107\,928\,278\,317 + 9\,922\,782\,870 n$

for $n = 0, 1, \ldots, 16$.

Proof
First we note that:


 * $107\,928\,278\,317 - 9\,922\,782\,870 = 98\,005\,495\,447 = 29 \times 149 \times 22\,681\,207$

and so this arithmetic progression of primes does not extend to $n < 0$.

But note that $107\,928\,278\,317 + 18 \times 9\,922\,782\,870 = 286\,538\,369\,977 = 23 \times 181 \times 68\,829\,779$ and so is not prime.