# Smallest 5 Consecutive Primes in Arithmetic Progression

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## Theorem

The smallest $5$ consecutive primes in arithmetic progression are:

- $9 \, 843 \, 019 + 30 n$

for $n = 0, 1, 2, 3, 4$.

Note that while there are many longer arithmetic progressions of far smaller primes, those primes are not consecutive.

## Proof

\(\displaystyle 9 \, 843 \, 019 + 0 \times 30\) | \(=\) | \(\displaystyle 9 \, 843 \, 019\) | which is the $654 \, 926$th prime | ||||||||||

\(\displaystyle 9 \, 843 \, 019 + 1 \times 30\) | \(=\) | \(\displaystyle 9 \, 843 \, 049\) | which is the $654 \, 927$th prime | ||||||||||

\(\displaystyle 9 \, 843 \, 019 + 2 \times 30\) | \(=\) | \(\displaystyle 9 \, 843 \, 079\) | which is the $654 \, 928$th prime | ||||||||||

\(\displaystyle 9 \, 843 \, 019 + 3 \times 30\) | \(=\) | \(\displaystyle 9 \, 843 \, 109\) | which is the $654 \, 929$th prime | ||||||||||

\(\displaystyle 9 \, 843 \, 019 + 4 \times 30\) | \(=\) | \(\displaystyle 9 \, 843 \, 139\) | which is the $654 \, 930$th prime |

But note that $9 \, 843 \, 019 + 5 \times 30 = 9 \, 843 \, 169 = 7^2 \times 200 \, 881$ and so is not prime.

Inspection of tables of primes (or a computer search) will reveal that this is the smallest such sequence.

$\blacksquare$

## Sources

- Jan. 1967: M.F. Jones, M. Lal and W.J. Blundon:
*Statistics on Certain Large Primes*(*Math. Comp.***Vol. 21**,*no. 97*: 103 – 107) www.jstor.org/stable/2003476

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $9,843,019$