Definition:Minimal Prime

Definition

Let $b \in \N_{>1}$ be a natural number.

A minimal prime base $b$ is a prime number whose base-$b$ representation has no proper subsequence which is also a prime number.

If $b$ is not specified, then decimal notation (that is, base $10$) is assumed.

Sequence

The complete sequence of minimal primes base $10$ is:

$2$, $3$, $5$, $7$, $11$, $19$, $41$, $61$, $89$, $409$, $449$, $499$, $881$, $991$, $6469$, $6949$, $9001$, $9049$, $9649$, $9949$, $60 \, 649$, $666 \, 649$, $946 \, 669$, $60 \, 000 \, 049$, $66 \, 000 \, 049$, $66 \, 600 \, 049$

Examples

$881$ is a minimal prime in base $10$ because there is no prime among the proper subsequences of the digits: $8$, $1$, $88$, $81$.

The subsequence does not have to consist of consecutive digits.

Hence $149$ is not a minimal prime in base $10$, because $19$ is prime.

However, by the definition of subsequence, the digits do have to be in the same order.

So, for example, $991$ is still a minimal prime in base $10$, even though a subset of the digits can form the shorter prime $19$ by changing the order.

Also see

• Results about minimal primes can be found here.

Sources

• 1999 -- 2000: J. Shallit: Minimal primes (J. Recr. Math. Vol. 30, no. 2: pp. 113 – 117)