Decimal truncatable primes
There are 4260 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, ... 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 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 920,720,315 left-and-right-truncatable primes: :2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733, 739, 751, 757, 773, 821, 823, 827, 829, 839, 853, 857, 859, 877, 929, 937, 953, 971, 977, 1117, 1171, 1193, 1231, 1237, 1291, 1297, 1319, 1373, 1433, 1439, 1471, 1531, 1597, 1613, 1619, ... There are 331,780,864 left-and-right-truncatable primes with an odd number of digits. The largest is the 97-digit prime 7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177. There are 588,939,451 left-and-right-truncatable primes with an even number of digits. The largest is the 104-digit prime 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973. 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 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, ... 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 ==Other bases==