There are 83 Right-Truncatable Primes in Base 10

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Theorem

In base $10$, there are $83$ right-truncatable primes:

$2$, $3$, $5$, $7$,
$23$, $29$, $31$, $37$, $53$, $59$, $71$, $73$, $79$,
$233$, $239$, $293$, $311$, $313$, $317$, $373$, $379$, $593$, $599$, $719$, $733$, $739$, $797$,
$2333$, $2339$, $2393$, $2399$, $2939$, $3119$, $3137$, $3733$, $3739$, $3793$, $3797$, $5939$, $7193$, $7331$, $7333$, $7393$,
$23 \, 333$, $23 \, 339$, $23 \, 399$, $23 \, 993$, $29 \, 399$, $31 \, 193$, $31 \, 379$, $37 \, 337$, $37 \, 339$, $37 \, 397$, $59 \, 393$, $59 \, 399$, $71 \, 933$, $73 \, 331$, $73 \, 939$,
$233 \, 993$, $239 \, 933$, $293 \, 999$, $373 \, 379$, $373 \, 393$, $593 \, 933$, $593 \, 993$, $719 \, 333$, $739 \, 391$, $739 \, 393$, $739 \, 397$, $739 \, 399$,
$2 \, 339 \, 933$, $2 \, 399 \, 333$, $2 \, 939 \, 999$, $3 \, 733 \, 799$, $5 \, 939 \, 333$, $7 \, 393 \, 913$, $7 \, 393 \, 931$, $7 \, 393 \, 933$,
$23 \, 399 \, 339$, $29 \, 399 \, 999$, $37 \, 337 \, 999$, $59 \, 393 \, 339$, $73 \, 939 \, 133$


Proof

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


Of the $2$-digit numbers starting with $2$, only $23$ and $29$ are primes.

Of the $2$-digit numbers starting with $3$, only $31$ and $37$ are primes.

Of the $2$-digit numbers starting with $5$, only $53$ and $59$ are primes.

Of the $2$-digit numbers starting with $7$, only $71$, $73$ and $79$ are primes.


Of the $3$-digit numbers starting with $23$, only $233$ and $239$ are primes.

Of the $3$-digit numbers starting with $29$, only $293$ is prime.

Of the $3$-digit numbers starting with $31$, only $311$, $313$ and $317$ are primes.

Of the $3$-digit numbers starting with $37$, only $373$ and $379$ are primes.

Of the $3$-digit numbers starting with $53$, none are prime.

Of the $3$-digit numbers starting with $59$, only $593$ and $599$ are primes.

Of the $3$-digit numbers starting with $71$, only $719$ is prime.

Of the $3$-digit numbers starting with $73$, only $733$ and $739$ are primes.

Of the $3$-digit numbers starting with $79$, only $797$ is prime.


Of the $4$-digit numbers starting with $233$, only $2333$ and $2339$ are primes.

Of the $4$-digit numbers starting with $239$, only $2393$ and $2399$ are primes.

Of the $4$-digit numbers starting with $293$, only $2939$ is prime.

Of the $4$-digit numbers starting with $311$, only $3119$ is prime.

Of the $4$-digit numbers starting with $313$, only $3137$ is prime.

Of the $4$-digit numbers starting with $317$, none are prime.

Of the $4$-digit numbers starting with $373$, only $3733$ and $3739$ are primes.

Of the $4$-digit numbers starting with $379$, only $3793$ and $3797$ are primes.

Of the $4$-digit numbers starting with $593$, only $5939$ is prime.

Of the $4$-digit numbers starting with $599$, none are prime.

Of the $4$-digit numbers starting with $719$, only $7193$ is prime.

Of the $4$-digit numbers starting with $733$, only $7331$ and $7333$ are primes.

Of the $4$-digit numbers starting with $739$, only $7393$ is prime.

Of the $4$-digit numbers starting with $797$, none are prime.


Of the $5$-digit numbers starting with $2333$, only $23 \, 333$ and $23 \, 339$ are primes.

Of the $5$-digit numbers starting with $2339$, only $23 \, 399$ is prime.

Of the $5$-digit numbers starting with $2393$, none are prime.

Of the $5$-digit numbers starting with $2399$, only $23 \, 993$ is prime.

Of the $5$-digit numbers starting with $2939$, only $29 \, 399$ is prime.

Of the $5$-digit numbers starting with $3119$, only $31 \, 193$ is prime.

Of the $5$-digit numbers starting with $3137$, only $31 \, 379$ is prime.

Of the $5$-digit numbers starting with $3733$, only $37 \, 337$ and $37 \, 339$ are primes.

Of the $5$-digit numbers starting with $3739$, only $37 \, 397$ is prime.

Of the $5$-digit numbers starting with $3793$, none are prime.

Of the $5$-digit numbers starting with $3797$, none are prime.

Of the $5$-digit numbers starting with $5939$, only $59 \, 393$ and $59 \, 399$ are primes.

Of the $5$-digit numbers starting with $7193$, only $71 \, 933$ is prime.

Of the $5$-digit numbers starting with $7331$, none are prime.

Of the $5$-digit numbers starting with $7333$, only $73 \, 331$ is prime.

Of the $5$-digit numbers starting with $7393$, only $73 \, 939$ is prime.


Of the $6$-digit numbers starting with $23 \, 333$, none are prime.

Of the $6$-digit numbers starting with $23 \, 339$, none are prime.

Of the $6$-digit numbers starting with $23 \, 399$, only $233 \, 993$ is prime.

Of the $6$-digit numbers starting with $23 \, 993$, only $239 \, 933$ is prime.

Of the $6$-digit numbers starting with $29 \, 399$, only $293 \, 999$ is prime.

Of the $6$-digit numbers starting with $31 \, 193$, none are prime.

Of the $6$-digit numbers starting with $31 \, 379$, none are prime.

Of the $6$-digit numbers starting with $37 \, 337$, only $373 \, 379$ is prime.

Of the $6$-digit numbers starting with $37 \, 339$, only $373 \, 393$ is prime.

Of the $6$-digit numbers starting with $37 \, 397$, none are prime.

Of the $6$-digit numbers starting with $59 \, 393$, only $593 \, 933$ is prime.

Of the $6$-digit numbers starting with $59 \, 399$, only $593 \, 993$ is prime.

Of the $6$-digit numbers starting with $71 \, 933$, only $719 \, 333$ is prime.

Of the $6$-digit numbers starting with $73 \, 331$, none are prime.

Of the $6$-digit numbers starting with $73 \, 939$, only $739 \, 391$, $739 \, 393$, $739 \, 397$ and $739 \, 399$ are primes.


Of the $7$-digit numbers starting with $233 \, 993$, only $2 \, 339 \, 933$ is prime.

Of the $7$-digit numbers starting with $239 \, 933$, only $2 \, 399 \, 333$ is prime.

Of the $7$-digit numbers starting with $293 \, 999$, only $2 \, 939 \, 999$ is prime.

Of the $7$-digit numbers starting with $373 \, 379$, only $3 \, 733 \, 799$ is prime.

Of the $7$-digit numbers starting with $373 \, 393$, none are prime.

Of the $7$-digit numbers starting with $593 \, 933$, only $5 \, 939 \, 333$ is prime.

Of the $7$-digit numbers starting with $593 \, 993$, none are prime.

Of the $7$-digit numbers starting with $719 \, 333$, none are prime.

Of the $7$-digit numbers starting with $739 \, 391$, only $7 \, 393 \, 913$ is prime.

Of the $7$-digit numbers starting with $739 \, 393$, only $7 \, 393 \, 931$ and $7 \, 393 \, 933$ are primes.

Of the $7$-digit numbers starting with $739 \, 397$, none are prime.

Of the $7$-digit numbers starting with $739 \, 399$, none are prime.


Of the $8$-digit numbers starting with $2 \, 339 \, 933$, only $23 \, 399 \, 339$ is prime.

Of the $8$-digit numbers starting with $2 \, 399 \, 333$, none are prime.

Of the $8$-digit numbers starting with $2 \, 939 \, 999$, only $29 \, 399 \, 999$ is prime.

Of the $8$-digit numbers starting with $3 \, 733 \, 799$, only $37 \, 337 \, 999$ is prime.

Of the $8$-digit numbers starting with $5 \, 939 \, 333$, only $59 \, 393 \, 339$ is prime.

Of the $8$-digit numbers starting with $7 \, 393 \, 913$, only $73 \, 939 \, 133$ is prime.

Of the $8$-digit numbers starting with $7 \, 393 \, 931$, none of them is prime.

Of the $8$-digit numbers starting with $7 \, 393 \, 933$, none of them is prime.


Of the $9$-digit numbers starting with $23 \, 399 \, 339$, none of them is prime.

Of the $9$-digit numbers starting with $29 \, 399 \, 999$, none of them is prime.

Of the $9$-digit numbers starting with $37 \, 337 \, 999$, none of them is prime.

Of the $9$-digit numbers starting with $59 \, 393 \, 339$, none of them is prime.

Of the $9$-digit numbers starting with $73 \, 939 \, 133$, none of them is prime.

$\blacksquare$

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