Truncatable prime
Encyclopedia
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. For example 7393, since 7393, 739, 73, 7 are all prime.
There are exactly 4260 decimal left-truncatable primes:
The largest is the 24-digit 357686312646216567629137.
There are 83 right-truncatable primes. The complete list:
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:
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.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a left-truncatable prime is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
which, in a given base
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
, 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
Decimal
The decimal numeral system has ten as its base. It is the numerical base most widely used by modern civilizations....
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. For example 7393, since 7393, 739, 73, 7 are all prime.
There are exactly 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, 1223, 1283, 1367 ...
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 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
While the primality of a number does not depend on the numeral system
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
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.