# Smallest 18 Primes in Arithmetic Progression

## 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 $n < 0$.

 $\displaystyle 107\,928\,278\,317 + 0 \times 9\,922\,782\,870$ $=$ $\displaystyle 107\,928\,278\,317$ which is prime $\displaystyle 107\,928\,278\,317 + 1 \times 9\,922\,782\,870$ $=$ $\displaystyle 117\,851\,061\,187$ which is prime $\displaystyle 107\,928\,278\,317 + 2 \times 9\,922\,782\,870$ $=$ $\displaystyle 127\,773\,844\,057$ which is prime $\displaystyle 107\,928\,278\,317 + 3 \times 9\,922\,782\,870$ $=$ $\displaystyle 137\,696\,626\,927$ which is prime $\displaystyle 107\,928\,278\,317 + 4 \times 9\,922\,782\,870$ $=$ $\displaystyle 147\,619\,409\,797$ which is prime $\displaystyle 107\,928\,278\,317 + 5 \times 9\,922\,782\,870$ $=$ $\displaystyle 157\,542\,192\,667$ which is prime $\displaystyle 107\,928\,278\,317 + 6 \times 9\,922\,782\,870$ $=$ $\displaystyle 167\,464\,975\,537$ which is prime $\displaystyle 107\,928\,278\,317 + 7 \times 9\,922\,782\,870$ $=$ $\displaystyle 177\,387\,758\,407$ which is prime $\displaystyle 107\,928\,278\,317 + 8 \times 9\,922\,782\,870$ $=$ $\displaystyle 187\,310\,541\,277$ which is prime $\displaystyle 107\,928\,278\,317 + 9 \times 9\,922\,782\,870$ $=$ $\displaystyle 197\,233\,324\,147$ which is prime $\displaystyle 107\,928\,278\,317 + 10 \times 9\,922\,782\,870$ $=$ $\displaystyle 207\,156\,107\,017$ which is prime $\displaystyle 107\,928\,278\,317 + 11 \times 9\,922\,782\,870$ $=$ $\displaystyle 217\,078\,889\,887$ which is prime $\displaystyle 107\,928\,278\,317 + 12 \times 9\,922\,782\,870$ $=$ $\displaystyle 227\,001\,672\,757$ which is prime $\displaystyle 107\,928\,278\,317 + 13 \times 9\,922\,782\,870$ $=$ $\displaystyle 236\,924\,455\,627$ which is prime $\displaystyle 107\,928\,278\,317 + 14 \times 9\,922\,782\,870$ $=$ $\displaystyle 246\,847\,238\,497$ which is prime $\displaystyle 107\,928\,278\,317 + 15 \times 9\,922\,782\,870$ $=$ $\displaystyle 256\,770\,021\,367$ which is prime $\displaystyle 107\,928\,278\,317 + 16 \times 9\,922\,782\,870$ $=$ $\displaystyle 266\,692\,804\,237$ which is prime $\displaystyle 107\,928\,278\,317 + 17 \times 9\,922\,782\,870$ $=$ $\displaystyle 276\,615\,587\,107$ which is prime

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.

## Historical Note

David Wells, in his Curious and Interesting Numbers of $1986$ reported that this was the longest known arithmetic progression of prime numbers.

As Paul A. Pritchard put it:

breaking the previous record of $17$ due to Weintraub.

The Weintraub in question was Sol Weintraub, who had discovered the previous record of $17$ such primes in $1977$.

Since that time, plenty longer have been found.