Smallest 17 Primes in Arithmetic Sequence

Theorem
The smallest $17$ primes in arithmetic sequence are:
 * $3\,430\,751\,869 + 87\,297\,210 n$

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

Proof
First we note that:


 * $3\,430\,751\,869 - 87\,297\,210 = 3\,343,\454\,659 = 17\,203 \times 194\,353$

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

But note that $3\,430\,751\,869 + 17 \times 87\,297\,210 = 4\,914\,804\,439 = 41 \times 97 \times 1 235807$ and so is not prime.