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).
There are 4260 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 920,720,315 left-and-right-truncatable primes: [1]
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:
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:
Similarly, a right-truncatable prime is called restricted if all of its right extensions are composite. There are 27 restricted right-truncatable primes:
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.
15 (fifteen) is the natural number following 14 and preceding 16.
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers.
In number theory, cousin primes are prime numbers that differ by four. Compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six.
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6.
79 (seventy-nine) is the natural number following 78 and preceding 80.
73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.
37 (thirty-seven) is the natural number following 36 and preceding 38.
In mathematics, a palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:
300 is the natural number following 299 and preceding 301.
700 is the natural number following 699 and preceding 701.
An emirp is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversible prime is used to mean the same as emirp, but may also, ambiguously, include the palindromic primes.
3000 is the natural number following 2999 and preceding 3001. It is the smallest number requiring thirteen letters in English.
7000 is the natural number following 6999 and preceding 7001.
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are
An undulating number is a number that has the digit form ABABAB... when in the base 10 number system. It is sometimes restricted to non-trivial undulating numbers which are required to have at least three digits and A ≠ B. The first few such numbers are:
In number theory, a full reptend prime, full repetend prime, proper prime or long prime in base b is an odd prime number p such that the Fermat quotient
189 is the natural number following 188 and preceding 190.
Super-prime numbers, also known as higher-order primes or prime-indexed primes (PIPs), are the subsequence of prime numbers that occupy prime-numbered positions within the sequence of all prime numbers.
A repeating decimal or recurring decimal is decimal representation of a number whose digits are periodic and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating. For example, the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is 3227/555, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals.