Permutable prime
Encyclopedia
A permutable prime is a prime number
, which, in a given base, can have its digits' positions switched through any permutation
and still spell a prime number. H. E. Richert, who supposedly first studied these primes, called them permutable primes, but later they were also called absolute primes.
In base 10, all the permutable primes with less than 49081 digits are :
where Rn =
is the number with n ones.
Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.
All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proved that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.
There is no n-digit permutable prime for 3 < n < 6·10175 which is not a repunit. It is conjecture
d that there are no non-repunit permutable primes other than those listed above.
In base 2, only repunits can be permutable primes, because any 0 permuted to the one's place results in an even number. Therefore the base 2 permutable primes are the Mersenne prime
s. The generalization can safely be made that for any positional number system, permutable primes with more than one digit can only have digits that are coprime
with the radix
of the number system. One-digit primes, meaning any prime below the radix, are always permutable.
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, can have its digits' positions switched through any permutation
Permutation
In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...
and still spell a prime number. H. E. Richert, who supposedly first studied these primes, called them permutable primes, but later they were also called absolute primes.
In base 10, all the permutable primes with less than 49081 digits are :
- 2, 3, 5, 7, 1111 (number)11 is the natural number following 10 and preceding 12.Eleven is the first number which cannot be counted with a human's eight fingers and two thumbs additively. In English, it is the smallest positive integer requiring three syllables and the largest prime number with a single-morpheme name...
, 1313 (number)13 is the natural number after 12 and before 14. It is the smallest number with eight letters in its name spelled out in English. It is also the first of the teens – the numbers 13 through 19 – the ages of teenagers....
, 1717 (number)17 is the natural number following 16 and preceding 18. It is prime.In spoken English, the numbers 17 and 70 are sometimes confused because they sound similar. When carefully enunciated, they differ in which syllable is stressed: 17 vs 70...
, 3131 (number)31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 3737 (number)37 is the natural number following 36 and preceding 38.-In mathematics:It is a prime number, the fifth lucky prime, the first irregular prime, the third unique prime and the third cuban prime of the form...
, 7171 (number)71 is the natural number following 70 and preceding 72.-In mathematics:71 is the algebraic degree of Conway's constant, a remarkable number arising in the study of look-and-say sequences....
, 7373 (number)73 is the natural number following 72 and preceding 74. In English, it is the smallest integer with twelve letters in its spelled out name.- In mathematics :...
, 7979 (number)Seventy-nine is the natural number following 78 and preceding 80.79 may represent:-In mathematics:*An odd number*The smallest number that can't be represented as a sum of fewer than 19 fourth powers*A strictly non-palindromic number...
, 9797 (number)97 is the natural number following 96 and preceding 98.-In mathematics:97 is the 25th prime number , following 89 and preceding 101. 97 is a Proth prime as it is 3 × 25 + 1.The numbers 97, 907, 9007, 90007 and 900007 are happy primes...
, 113113 (number)113 is the natural number following 112 and preceding 114.-In mathematics:One hundred [and] thirteen is the 30th prime number, following 109 and preceding 127, a Sophie Germain prime, a Chen prime and a Proth prime as it is a prime number of the form 7 × 24 + 1...
, 131131 (number)131 is the natural number following 130 and preceding 132.-In mathematics:131 is a Sophie Germain prime, the second 3-digit palindromic prime, and also a permutable prime with 113 and 311. It can be expressed as the sum of three consecutive primes, 131 = 41 + 43 + 47. 131 is an Eisenstein prime...
, 199199 (number)199 is the natural number following 198 and preceding 200.-In mathematics:* 199 is an odd number* 199 is a centered triangular number* 199 is a centered 33-gonal number* 199 is a deficient number, as 1 is less than 199* 199 is a Lucas number...
, 311, 337, 373, 733, 919, 991, 1111111111111111111, 11111111111111111111111, R317, R1031
where Rn =
is the number with n ones.Any repunit prime is a permutable prime with the above definition, but some definitions require at least two distinct digits.
All permutable primes of two or more digits are composed from the digits 1, 3, 7, 9, because no prime number except 2 is even, and no prime number besides 5 is divisible by 5. It is proved that no permutable prime exists which contains three different of the four digits 1, 3, 7, 9, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 3, 7, 9.
There is no n-digit permutable prime for 3 < n < 6·10175 which is not a repunit. It is conjecture
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
d that there are no non-repunit permutable primes other than those listed above.
In base 2, only repunits can be permutable primes, because any 0 permuted to the one's place results in an even number. Therefore the base 2 permutable primes are the Mersenne prime
Mersenne prime
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
s. The generalization can safely be made that for any positional number system, permutable primes with more than one digit can only have digits that are coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
with the radix
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...
of the number system. One-digit primes, meaning any prime below the radix, are always permutable.