Definition:Long Period Prime

Definition

A long period prime is a prime number $p$ whose reciprocal has the maximum period $p - 1$.

Sequence

The sequence of long period primes begins:

$7$, $17$, $19$, $23$, $29$, $47$, $59$, $61$, $97$, $109$, $113$, $131$, $149$, $167$, $179$, $\ldots$

Examples

$7$ is Long Period Prime

$7$ is the smallest long period prime:

$\dfrac 1 7 = 0 \cdotp \dot 14285 \dot 7$

$17$ is Long Period Prime

The prime number $17$ is a long period prime:

$\dfrac 1 {17} = 0 \cdotp \dot 05882 \, 35294 \, 11764 \, \dot 7$

$19$ is Long Period Prime

The prime number $19$ is a long period prime:

$\dfrac 1 {19} = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1$

$23$ is Long Period Prime

The prime number $23$ is a long period prime:

$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

$29$ is Long Period Prime

The prime number $29$ is a long period prime:

$\dfrac 1 {29} = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1$

$47$ is Long Period Prime

The prime number $47$ is a long period prime:

$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$

$59$ is Long Period Prime

The prime number $59$ is a long period prime:

$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$

$61$ is Long Period Prime

The prime number $61$ is a long period prime:

$\dfrac 1 {61} = 0 \cdotp \dot 01639 \, 34426 \, 22950 \, 81967 \, 21311 \, 47540 \, 98360 \, 65573 \, 77049 \, 18032 \, 78688 \, 5245 \dot 9$

Also see

• Results about long period primes can be found here.

Sources

• On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008): A001913