# 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.

## Contents

## History[edit]

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[edit]

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:

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[edit]

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 10^{n} digits must be at least 10^{n−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.

## See also[edit]

## References[edit]

- Weisstein, Eric W. "Truncatable Prime".
*MathWorld*. - Caldwell, Chris,
*left-truncatable prime*and*right-truncatable primes*, at the Prime Pages glossary. - Rivera, Carlos, Problems & Puzzles: Puzzle 2.- Prime strings and Puzzle 131.- Growing primes

## External Links[edit]

- Grime, Dr. James. "357686312646216567629137" (video).
*YouTube*. Brady Haran. Retrieved 27 July 2018.