# Definition:Two-Sided Prime

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## Definition

A **two-sided prime** is a prime number which remains prime when:

- any number of digits are removed from the left hand end

and:

- any number of digits are removed from the right hand end

but, generally, not from both ends at once.

### Sequence

The complete sequence of two-sided primes is:

- $2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739 \, 397$

## Examples

### $739 \, 397$ is a Two-Sided Prime

\(\displaystyle 739 \, 397\) | \(\) | \(\displaystyle \) | is the $59 \, 489$th prime | ||||||||||

\(\displaystyle 39 \, 397\) | \(\) | \(\displaystyle \) | is the $4148$th prime | ||||||||||

\(\displaystyle 9397\) | \(\) | \(\displaystyle \) | is the $1162$nd prime | ||||||||||

\(\displaystyle 397\) | \(\) | \(\displaystyle \) | is the $75$th prime | ||||||||||

\(\displaystyle 97\) | \(\) | \(\displaystyle \) | is the $25$th prime | ||||||||||

\(\displaystyle 7\) | \(\) | \(\displaystyle \) | is the $4$th prime |

\(\displaystyle \) | \(\) | \(\displaystyle 739 \, 397\) | is the $59 \, 489$th prime | ||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle 73 \, 939\) | is the $7296$th prime | ||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle 7393\) | is the $939$th prime | ||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle 739\) | is the $131$st prime | ||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle 73\) | is the $21$st prime | ||||||||||

\(\displaystyle \) | \(\) | \(\displaystyle 7\) | is the $4$th prime |

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $739,397$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $739,397$