Truncatable prime

Type of number
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, and 7 are all prime. A left and right truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one or two digit prime. 1825711 is an example of a left and right truncatable prime, since 1825711, 82571, 257, and 5 are all prime. In base 10, there are exactly 4260 left truncatable primes, 83 right truncatable primes, and 920,720,315 left and right truncatable primes.

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).

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.