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Truncatable Primes ( Trivia # 19 )

on Tuesday, May 12, 2009

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 ...

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


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.

4 comments:

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