In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime.

For example 9137, since 9137, 137, 37 and 7 are all prime.

There are exactly 4260 decimal 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, 1283, 1367 ...

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, 1283, 1367 ...

The largest left-truncatable prime is the 24-digit 357686312646216567629137.

A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed.

For example 7393, since 7393, 739, 73, 7 are all prime.

There are 83 right-truncatable primes. The complete list:

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, 23333, 23339,

23399, 23993, 29399, 31193, 31379,

37337, 37339, 37397, 59393, 59399,

71933, 73331, 73939, 233993, 239933,

293999, 373379, 373393, 593933, 593993,

719333, 739391, 739393, 739397, 739399,

2339933, 2399333, 2939999, 3733799, 5939333,

7393913, 7393931, 7393933, 23399339,

29399999, 37337999, 59393339, 73939133

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, 23333, 23339,

23399, 23993, 29399, 31193, 31379,

37337, 37339, 37397, 59393, 59399,

71933, 73331, 73939, 233993, 239933,

293999, 373379, 373393, 593933, 593993,

719333, 739391, 739393, 739397, 739399,

2339933, 2399333, 2939999, 3733799, 5939333,

7393913, 7393931, 7393933, 23399339,

29399999, 37337999, 59393339, 73939133

The prime 73939133 is the largest right-truncatable prime that repeatedly produces primes when digits are deleted from the right.

The numbers 73939133, 7393913, 739391, 73939, 7393, 739, 73, and 7 are all prime.

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