# There are 4260 Left-Truncatable Primes in Base 10

## Theorem

In base $10$, there are $4260$ left-truncatable primes:

$2$, $3$, $5$, $7$, $13$, $17$, $23$, $37$, $43$, $47$, $53$, $67$, $73$, $83$, $97$, $113$, $137$, $167$, $173$, $197$, $223$, $283$, $313$, $317$, $337$, $347$, $353$, $367$, $373$, $383$, $397$, $443$, $467$, $523$, $547$, $613$, $617$, $643$, $647$, $653$, $673$, $683$, $743$, $773$, $797$, $823$, $853$, $883$, $937$, $947$, $953$, $967$, $983$, $997$, $1223$, $\ldots$, $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$

## Proof

Of the $1$-digit numbers, only $2$, $3$, $5$, $7$ are primes.

Of the $2$-digit numbers ending with $2$, none are prime.

Of the $2$-digit numbers ending with $3$, $13$, $23$, $43$, $53$, $73$, and $83$ are primes.

Of the $2$-digit numbers ending with $5$, none are prime.

Of the $2$-digit numbers ending with $7$, $17$, $37$, $47$, $67$, and $97$ are primes.

Of the $3$-digit numbers ending with $13$, $113$, $313$, $613$ are primes.

...

Of the $24$-digit numbers ending with $57 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$, only $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$ is prime.

Of the $25$-digit numbers ending with $357 \, 686 \, 312 \, 646 \, 216 \, 567 \, 629 \, 137$, none are prime.

$\blacksquare$