Smallest 10 Primes in Arithmetic Progression

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Theorem

The smallest $10$ primes in arithmetic progression are:

$199 + 210 n$

for $n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9$.

These are also the smallest $8$ and $9$ primes in arithmetic progression.


Proof

\(\displaystyle 199 + 0 \times 210\) \(=\) \(\displaystyle 199\) which is the $46$th prime
\(\displaystyle 199 + 1 \times 210\) \(=\) \(\displaystyle 409\) which is the $80$th prime
\(\displaystyle 199 + 2 \times 210\) \(=\) \(\displaystyle 619\) which is the $114$th prime
\(\displaystyle 199 + 3 \times 210\) \(=\) \(\displaystyle 829\) which is the $145$th prime
\(\displaystyle 199 + 4 \times 210\) \(=\) \(\displaystyle 1039\) which is the $175$th prime
\(\displaystyle 199 + 5 \times 210\) \(=\) \(\displaystyle 1249\) which is the $204$th prime
\(\displaystyle 199 + 6 \times 210\) \(=\) \(\displaystyle 1459\) which is the $232$nd prime
\(\displaystyle 199 + 7 \times 210\) \(=\) \(\displaystyle 1669\) which is the $263$rd prime
\(\displaystyle 199 + 8 \times 210\) \(=\) \(\displaystyle 1879\) which is the $289$th prime
\(\displaystyle 199 + 9 \times 210\) \(=\) \(\displaystyle 2089\) which is the $316$th prime

This sequence is A033168 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


But note that $199 + 10 \times 210 = 2299 = 11^2 \times 19$ and so is not prime.



Sources