# Smallest 18 Primes in Arithmetic Sequence

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## Theorem

The smallest $18$ primes in arithmetic sequence 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 sequence of primes does not extend to $n < 0$.

\(\ds 107\,928\,278\,317 + 0 \times 9\,922\,782\,870\) | \(=\) | \(\ds 107\,928\,278\,317\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 1 \times 9\,922\,782\,870\) | \(=\) | \(\ds 117\,851\,061\,187\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 2 \times 9\,922\,782\,870\) | \(=\) | \(\ds 127\,773\,844\,057\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 3 \times 9\,922\,782\,870\) | \(=\) | \(\ds 137\,696\,626\,927\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 4 \times 9\,922\,782\,870\) | \(=\) | \(\ds 147\,619\,409\,797\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 5 \times 9\,922\,782\,870\) | \(=\) | \(\ds 157\,542\,192\,667\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 6 \times 9\,922\,782\,870\) | \(=\) | \(\ds 167\,464\,975\,537\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 7 \times 9\,922\,782\,870\) | \(=\) | \(\ds 177\,387\,758\,407\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 8 \times 9\,922\,782\,870\) | \(=\) | \(\ds 187\,310\,541\,277\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 9 \times 9\,922\,782\,870\) | \(=\) | \(\ds 197\,233\,324\,147\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 10 \times 9\,922\,782\,870\) | \(=\) | \(\ds 207\,156\,107\,017\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 11 \times 9\,922\,782\,870\) | \(=\) | \(\ds 217\,078\,889\,887\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 12 \times 9\,922\,782\,870\) | \(=\) | \(\ds 227\,001\,672\,757\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 13 \times 9\,922\,782\,870\) | \(=\) | \(\ds 236\,924\,455\,627\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 14 \times 9\,922\,782\,870\) | \(=\) | \(\ds 246\,847\,238\,497\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 15 \times 9\,922\,782\,870\) | \(=\) | \(\ds 256\,770\,021\,367\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 16 \times 9\,922\,782\,870\) | \(=\) | \(\ds 266\,692\,804\,237\) | which is prime | |||||||||||

\(\ds 107\,928\,278\,317 + 17 \times 9\,922\,782\,870\) | \(=\) | \(\ds 276\,615\,587\,107\) | 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.

This theorem requires a proof.In particular: It remains to be shown that there are no smaller such APsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Historical Note

David Wells, in his *Curious and Interesting Numbers* of $1986$ reported that this was the longest known arithmetic sequence 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

- 1983: Paul A. Pritchard:
*Eighteen Primes in Arithmetic Progression*(*Math. Comp.***Vol. 41**: p. 697) www.jstor.org/stable/2007705 - 1986: David Wells:
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