Smallest 18 Primes in Arithmetic Progression

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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.


Sources