# Longest Sequence of Consecutive Primes in Arithmetic Sequence

## Sequence

The longest known sequence of consecutive prime numbers in arithmetic sequence starts at $121 \, 174 \, 811$, has a length of $6$ and a common difference of $30$:

$121 \, 174 \, 811, 121 \, 174 \, 841, 121 \, 174 \, 871, 121 \, 174 \, 901, 121 \, 174 \, 931, 121 \, 174 \, 961$

## Proof

 $\ds 121 \, 174 \, 811 + 0 \times 30$ $=$ $\ds 121 \, 174 \, 811$ which is the $6 \, 904 \, 737$th prime $\ds 121 \, 174 \, 811 + 1 \times 30$ $=$ $\ds 121 \, 174 \, 841$ which is the $6 \, 904 \, 738$th prime $\ds 121 \, 174 \, 811 + 2 \times 30$ $=$ $\ds 121 \, 174 \, 871$ which is the $6 \, 904 \, 739$th prime $\ds 121 \, 174 \, 811 + 3 \times 30$ $=$ $\ds 121 \, 174 \, 901$ which is the $6 \, 904 \, 740$th prime $\ds 121 \, 174 \, 811 + 4 \times 30$ $=$ $\ds 121 \, 174 \, 931$ which is the $6 \, 904 \, 741$st prime $\ds 121 \, 174 \, 811 + 5 \times 30$ $=$ $\ds 121 \, 174 \, 961$ which is the $6 \, 904 \, 742$nd prime

But note that $121 \, 174 \, 811 + 6 \times 30 = 121 \, 174 \, 991 = 7^2 \times 2 \, 472 \, 959$ and so is not prime.

Inspection of tables of primes (or a computer search) will reveal that no longer such sequences are known.

$\blacksquare$