# Smallest 10 Primes in Arithmetic Progression

## 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

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