Truncatable prime

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. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, 7 are all prime.

There are 4260 left-truncatable primes and 83 right-truncatable primes.

History

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 November 1969 issue of Mathematics Magazine, where truncatable primes were called prime primes by two co-authors (Murray Berg and John E. Walstrom).

Lists of truncatable primes

There are known to be 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 the 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 the 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 the 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 7937 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7937 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 A240768 in the 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 the OEIS)

Other bases

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.

The left-truncatable primes in base 12 are: (using inverted two and three for ten and eleven, respectively)

2, 3, 5, 7, Ɛ, 15, 17, 1Ɛ, 25, 27, 35, 37, 3Ɛ, 45, 4Ɛ, 57, 5Ɛ, 67, 6Ɛ, 75, 85, 87, 8Ɛ, 95, ᘔ7, ᘔƐ, Ɛ5, Ɛ7, 117, 11Ɛ, 125, 13Ɛ, 145, 157, 167, 16Ɛ, 175, 18Ɛ, 195, 1ᘔ7, 1Ɛ5, 1Ɛ7, 217, 21Ɛ, 225, 237, 24Ɛ, 25Ɛ, 267, 285, 295, 2ᘔƐ, 315, 325, 327, 33Ɛ, 34Ɛ, 357, 35Ɛ, 375, 3ᘔƐ, 3Ɛ5, 3Ɛ7, 415, 41Ɛ, 427, 435, 437, 457, 45Ɛ, 46Ɛ, 485, 48Ɛ, 517, 51Ɛ, 527, 535, 545, 557, 575, 585, 587, 58Ɛ, 5Ɛ5, 5Ɛ7, 615, 617, 61Ɛ, 637, 63Ɛ, 66Ɛ, 675, 687, 68Ɛ, 695, 6ᘔ7, 71Ɛ, 727, 735, 737, 745, 767, 76Ɛ, 775, 785, 817, 825, 835, 85Ɛ, 867, 88Ɛ, 8ᘔ7, 8ᘔƐ, 8Ɛ5, 8Ɛ7, 91Ɛ, 927, 95Ɛ, 987, 995, 9ᘔ7, 9ᘔƐ, 9Ɛ5, ᘔ17, ᘔ27, ᘔ35, ᘔ37, ᘔ3Ɛ, ᘔ45, ᘔ4Ɛ, ᘔ5Ɛ, ᘔ6Ɛ, ᘔ87, ᘔ95, ᘔᘔ7, ᘔᘔƐ, ᘔƐ7, Ɛ15, Ɛ1Ɛ, Ɛ25, Ɛ37, Ɛ45, Ɛ67, Ɛ6Ɛ, Ɛ95, ƐƐ5, ƐƐ7, ...

There are 170053 left-truncatable primes in base 12, the largest is the 32-digit 471ᘔ34ᘔ164259Ɛᘔ16Ɛ324ᘔƐ8ᘔ32Ɛ7817.

The right-truncatable primes in base 12 are: (using inverted two and three for ten and eleven, respectively)

2, 3, 5, 7, Ɛ, 25, 27, 31, 35, 37, 3Ɛ, 51, 57, 5Ɛ, 75, Ɛ5, Ɛ7, 251, 255, 25Ɛ, 271, 277, 27Ɛ, 315, 357, 35Ɛ, 375, 377, 3Ɛ5, 3Ɛ7, 511, 517, 51Ɛ, 575, 577, 5Ɛ1, 5Ɛ5, 5Ɛ7, 5ƐƐ, 751, Ɛ71, 2555, 2557, 2715, 2717, 2771, 27Ɛ1, 27Ɛ7, 3155, 315Ɛ, 35Ɛ1, 35Ɛ7, 35ƐƐ, 3755, 375Ɛ, 3771, 377Ɛ, 3Ɛ51, 3Ɛ55, 3Ɛ75, 3Ɛ7Ɛ, 5117, 511Ɛ, 51Ɛ7, 575Ɛ, 5771, 5777, 577Ɛ, 5Ɛ17, 5Ɛ1Ɛ, 5Ɛ55, 5Ɛ75, 5ƐƐ1, 7511, Ɛ711, 25551, 25577, 27151, 27155, 2715Ɛ, 27Ɛ17, 27Ɛ77, 31551, 315Ɛ5, 375Ɛ5, 375ƐƐ, 37715, 3Ɛ515, 3Ɛ557, 3Ɛ55Ɛ, 3Ɛ7Ɛ5, 511Ɛ7, 51Ɛ71, 575ƐƐ, 57711, 57717, 577Ɛ7, 577ƐƐ, 5Ɛ175, 5Ɛ1Ɛ7, 5Ɛ55Ɛ, 5Ɛ751, 5ƐƐ17, 75111, 75115, Ɛ7111, Ɛ7115, 255515, 255775, 271555, 2715Ɛ1, 27Ɛ177, 27Ɛ17Ɛ, 27Ɛ771, 375Ɛ55, 375ƐƐ5, 377151, 3Ɛ5155, 3Ɛ5157, 3Ɛ515Ɛ, 3Ɛ5571, 3Ɛ557Ɛ, 3Ɛ55Ɛ7, 3Ɛ7Ɛ5Ɛ, 511Ɛ77, 51Ɛ717, 575ƐƐƐ, 577117, 577175, 577Ɛ75, 5Ɛ55Ɛ1, 5Ɛ55ƐƐ, 5ƐƐ171, 751115, Ɛ71157, ...

A right-truncatable prime in base 12 can only contain digits {1, 5, 7, Ɛ} after the leading digit. There are 179 right-truncatable primes in base 12, the largest is the 10-digit 375ƐƐ5Ɛ515.