lefttruncatable

Lefttruncatable primes are primes that remain prime when the leftmost digit is removed, and this leftward truncation can be repeated until a single-digit prime is reached. In base 10, there are exactly 4260 such primes, including the one-digit primes 2, 3, 5, and 7 by convention. The set is finite in base 10, a property that does not hold for all prime-related truncations in every base.

Construction and properties: Starting from the one-digit primes, new left-truncatable primes are formed by prefixing digits and testing primality of the resulting number. A number is lefttruncatable if each successive left truncation yields a prime. For longer numbers, the final digit (the rightmost digit) must be one of 1, 3, 7, or 9 to preserve primality, so all nontrivial lefttruncatable primes end in one of these digits. The leftmost digit can be any nonzero digit, provided the full number is prime.

Largest example and scope: The largest lefttruncatable prime in base 10 is 357686312646216567629137, a 24-digit number. This, together with the 4 single-digit primes, accounts for the finite total of 4260 base-10 lefttruncatable primes. Similar families exist in other numerical bases, each with its own maximal length and total.

Context and related concepts: Lefttruncatable primes are a topic in recreational number theory and illustrate how truncation interacts with primality. They are related to righttruncatable primes and to primes that are truncatable from both ends, though those dual-truncation sets have different characteristics and sizes.