Definition:Unique Period Prime

Definition

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

A unique period prime base $b$ is a prime number $p$ whose reciprocal expressed base $b$ has a period shared with no other prime.

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

Sequence

The sequence of unique period primes begins:

$3$, $11$, $37$, $101$, $9091$, $9901$, $333 \, 667$, $909 \, 091$, $99 \, 990 \, 001$, $999 \, 999 \, 000 \, 001$, $9 \, 999 \, 999 \, 900 \, 000 \, 001$, $909 \, 090 \, 909 \, 090 \, 909 \, 091$, $1 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $11 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111 \, 111$, $900 \, 900 \, 900 \, 900 \, 990 \, 990 \, 990 \, 991$, $909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 090 \, 909 \, 091$, $\ldots$

Examples

In base $10$, the prime numbers $3$, $11$, $37$, and $101$ are the only ones with periods $1$, $2$, $3$ and $4$ respectively.

Hence they are unique period primes.

But:

$41$ and $271$ both have period $5$

$7$ and $13$ both have period $6$

$239$ and $4649$ both have period $7$

$73$ and $137$ both have period $8$.

Thus these primes are not unique period primes.

Also see

• Results about unique period primes can be found here.