Smallest 5 Consecutive Primes in Arithmetic Sequence

Theorem

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

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

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

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

$\ds 9 \, 843 \, 019 + 0 \times 30$

$=$

$\ds 9 \, 843 \, 019$

which is the $654 \, 926$th prime

$\ds 9 \, 843 \, 019 + 1 \times 30$

$=$

$\ds 9 \, 843 \, 049$

which is the $654 \, 927$th prime

$\ds 9 \, 843 \, 019 + 2 \times 30$

$=$

$\ds 9 \, 843 \, 079$

which is the $654 \, 928$th prime

$\ds 9 \, 843 \, 019 + 3 \times 30$

$=$

$\ds 9 \, 843 \, 109$

which is the $654 \, 929$th prime

$\ds 9 \, 843 \, 019 + 4 \times 30$

$=$

$\ds 9 \, 843 \, 139$

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.

$\blacksquare$

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

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