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