Smallest 22 Primes in Arithmetic Sequence

Theorem
The smallest $22$ primes in arithmetic progression are:
 * $11 \, 410 \, 337 \, 850 \, 553 + 4 \, 609 \, 098 \, 694 \, 200 n$

for $n = 0, 1, \ldots, 21$.

Proof
But note that $11 \, 410 \, 337 \, 850 \, 553 + 22 \times 4 \, 609 \, 098 \, 694 \, 200 = 112 \, 810 \, 509 \, 122 \, 953 = 61 \times 107 \times 1907 \times 9063277$ and so is not prime.