Truncatable prime

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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. Decimal representation is often assumed and always used in this article.

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 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, ... (sequence A024785 in OEIS)

The largest is the 24-digit 357686312646216567629137.

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 (sequence A024770 in OEIS)

The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit.

There are 15 primes which are both left-truncatable and right-truncatable. They have been called two-sided primes. The complete list:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence A020994 in OEIS)

A left-truncatable prime is called restricted if all of its left extensions are composite i.e. there is no other left-truncatable prime of which this prime is the left-truncated "tail". Thus 7939 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7939 are all composite, whereas 3797 is a left-truncatable prime that is not restricted because 33797 is also prime.

There are 1442 restricted left-truncatable primes:

2, 5, 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, 15647, 16823, 24373, 33547, 34337, 37643, 56983, 57853, 59743, 62383, 63347, 63617, 69337, 72467, 72617, 75653, 76367, 87643, 92683, 97883, 98317, ... (sequence A055521 in OEIS)

Similarly, a right-truncatable prime is called restricted if all of its right extensions are composite. There are 27 restricted right-truncatable primes:

53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence A239747 in OEIS)

While the primality of a number does not depend on the numeral system used, truncatable primes are defined only in relation with a given base. A variation involves removing 2 or more decimal digits at a time. This is mathematically equivalent to using base 100 or a larger power of 10, with the restriction that base 10n digits must be at least 10n−1, in order to match a decimal n-digit number with no leading 0.

An author named Leslie E. Card in early volumes of the Journal of Recreational Mathematics (which started its run in 1968) considered a topic close to that of right-truncatable primes, calling sequences that by adding digits to the right in sequence to an initial number not necessarily prime snowball primes.

Discussion of the topic dates to at least the November of 1969 issue of Mathematics Magazine, where truncatable primes were called prime primes by two co-authors (Murray Berg and John E. Walstrom).

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