truncatable primes

A prime

is called truncatable if all its prefixes or all its suffixes are primes. More precisely, they are respectively called right-truncatable primes, and left-truncatable primes.

In both cases only zeroless numbers are considered.

For example, the prime 23399 is a (right) truncatable prime because all the numbers 2, 23, 233, and 2339 are primes. Similarly, 26947 is a (left) truncatable prime because 7, 47, 947, and 6947 are primes.

There are 83, right-truncatable primes, the largest being 73939133, while the largest of the 4260 left-truncatable primes is the 24-digit prime 357686312646216567629137.

Clearly the concept of truncatability depends on the base in which the numbers are represented. For example, in base 100 (in which each single digit can be represented with two base 10 digits), there are exactly 9823399067 right-truncatable primes, the largest being 7 01 23 91 63 63 51 51 99 41 61 99 51 83 01 69 83 21 19 53 39 01 27 27 99 47 99 19 03 71 99 21 51 27 97 29 97 47 57 39 79 09 99 23 27 93 69 43 87 71 27 37 57 81 09 11 43.

The first left or right truncatable primes are 2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 113, 137, 167, 173, 197, 223, 233, 239, 283 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

Useful links Wikipedia, Truncatable prime
Mathworld, Truncatable Prime
OEIS, Sequence A024770
OEIS, Sequence A024785

Truncatable primes can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 293 3373 3643 3739 7333 + 9363243613 9459642683 aban 13 17 23 29 31 37 + 983 997 alternating 23 29 43 47 67 83 + 789616547 929636947 amenable 13 17 29 37 53 73 + 999636997 999818353 apocalyptic 443 647 823 937 983 1823 + 29173 29399 arithmetic 13 17 23 29 31 37 + 9981373 9986113 balanced p. 53 173 373 593 653 733 + 9127692647 9391564373 c.decagonal 31 c.heptagonal 43 71 197 547 953 2647 9283 c.pentagonal 31 c.square 13 113 313 613 2113 3613 + 86113 5661613 c.triangular 31 Carol 47 223 3967 Chen 13 17 23 29 31 37 + 99759467 99979337 congruent 13 23 29 31 37 47 + 9981373 9986113 constructible 17 Cunningham 17 31 37 197 3137 24337 Curzon 29 53 113 173 233 293 + 157573673 169956113 cyclic 13 17 23 29 31 37 + 9981373 9986113 d-powerful 43 283 373 379 739 2137 + 9566173 9979337 de Polignac 337 373 599 997 2683 3119 + 98966653 99336373 deficient 13 17 23 29 31 37 + 9981373 9986113 dig.balanced 37 197 613 617 647 653 + 186342467 198739397 economical 13 17 23 29 31 37 + 19861613 19981373 emirp 13 17 31 37 71 73 + 186342467 198739397 equidigital 13 17 23 29 31 37 + 19861613 19981373 esthetic 23 43 67 evil 17 23 29 43 53 71 + 999267523 999636997 fibodiv 47 Fibonacci 13 233 Friedman 347 132647 276673 279967 912647 Gilda 29 683 997 good prime 17 29 37 53 59 67 + 13294397 99537547 happy 13 23 31 79 97 167 + 9933613 9973547 hex 37 397 547 5167 93626947 Hogben 13 31 43 73 5113 Honaker 3137 4397 6353 6653 7283 7523 + 627543853 915613967 hungry 17 iban 17 23 43 47 71 73 + 12347 24373 iccanobiF 13 idoneal 13 37 inconsummate 173 383 443 983 3137 3733 + 969467 979337 Jacobsthal 43 683 junction 113 311 313 317 719 1613 + 98672953 99187547 katadrome 31 43 53 71 73 83 + 9743 87643 Kynea 23 79 Leyland 17 593 lonely 23 53 3967 Lucas 29 47 lucky 13 31 37 43 67 73 + 9891997 9981373 m-pointer 23 12113 magnanimous 23 29 43 47 67 83 + 683 4643 metadrome 13 17 23 29 37 47 + 3467 12347 modest 13 23 29 59 79 233 + 2939999 29399999 nialpdrome 31 43 53 71 73 83 + 998443 9866653 oban 13 17 23 29 37 53 + 983 997 odious 13 31 37 47 59 67 + 999818353 999962683 Ormiston 36997 37397 51613 56197 94397 454673 + 993946997 1543279337 palindromic 313 353 373 383 797 76367 + 7693967 799636997 palprime 313 353 373 383 797 76367 + 7693967 799636997 pancake 29 37 67 79 137 379 + 2347 8647 panconsummate 23 31 37 43 53 59 + 353 523 pernicious 13 17 31 37 47 59 + 9981373 9986113 Perrin 17 29 367 853 Pierpont 13 17 37 73 97 plaindrome 13 17 23 29 37 47 + 33347 233347 prime 13 17 23 29 31 37 + 997564326947 999631686353 primeval 13 37 113 137 1367 Proth 13 17 97 113 353 673 + 59393 986113 repfigit 47 197 repunit 13 31 43 73 5113 self 31 53 97 233 367 547 + 968666173 995729173 sliding 29 Sophie Germain 23 29 53 83 113 173 + 9391564373 9439663853 star 13 37 73 337 937 3313 + 243613 924337 strong prime 17 29 37 59 67 71 + 99818353 99951283 super-d 31 337 719 739 3119 3347 + 9891997 9973547 tetranacci 29 773 tribonacci 13 twin 13 17 29 31 43 59 + 993946997 998966653 uban 13 17 23 29 31 37 + 83 97 Ulam 13 47 53 97 197 673 + 9961613 9966883 undulating 313 353 373 383 797 upside-down 37 73 3467 7193 7283 7643 7823 weak prime 13 23 31 43 47 73 + 99962683 99979337 Woodall 17 23 383 4373