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