Permutable prime number

A permutable prime number (also known as an absolute prime number ) is a prime number in which any rearrangement of its digits also results in a prime number. For example, 113 is a permutable prime number because 131 and 311 are also prime. Whether this condition is met also depends on the value system used . When the mathematician Hans-Egon Richert first dealt with these numbers in an essay, he called them permutable prime numbers . Later authors also used the term absolute prime .

Permutable prime numbers in the decimal system

The first permutable prime numbers in the decimal system are the following:

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R ₁₉ (= 1111111111111111111) , R ₂₃ , R ₃₁₇ , R ₁₀₃₁ , R ₄₉₀₈₁ , R ₈₆₄₅₃ , R ₁₀₉₂₉₇ and R ₂₇₀₃₄₃ (sequence A003459 in OEIS )

Since the above list contains the numbers 113, 131 and 311, but can be converted into one another by permutating the digits, it makes sense to only specify the smallest number of this permutation class:

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, R ₁₉ , R ₂₃ , R ₃₁₇ , R ₁₀₃₁ , R ₄₉₀₈₁ , R ₈₆₄₅₃ , R ₁₀₉₂₉₇ and R ₂₇₀₃₄₃ (sequence A258706 in OEIS )

991 is the largest known permutable prime number that consists of different digits. All other known permutable prime numbers are repunits , i. H. Numbers that only contain the number 1. This has been proven for all n -digit numbers 3 < n <6 × 10 ¹⁷⁵ , but it is assumed that there are no other permutable prime numbers that consist of different digits. Accordingly, 2, 3, 5, 7, 13, 17, 37, 79, 113, 199, 337 and their permutations would be the only permutable prime numbers that are not successions (and according to an alternative definition, the different digits for two or more digits Numbers required, the only ones at all). The indices of the primary repunits can also be read off from sequence A004023 in OEIS . The most recent repunits and are PRP numbers , so it is not yet entirely certain whether they are really prime numbers. ${\displaystyle R_{49081},R_{86453},R_{109297}}$ ${\displaystyle R_{270343}}$

properties

• All multi-digit permutable prime numbers can necessarily only contain the digits 1, 3, 7 and 9, since the occurrence of an even number or a 5 would mean that at least one permutation would be divisible by 2 or by 5 and therefore not prime.
• A permutable prime number cannot contain all four of the above possible digits 1, 3, 7 and 9 at the same time; no permutable prime number is possible even with three different ones from these four digits.
• Only one of the maximum two different digits can appear twice or more than once.
• All permutable prime numbers that do not represent a prime number palindrome are also mirp numbers .

Permutable prime numbers with other bases

Examples

• The first permutable prime numbers with base 12 are the following (where A = 10 and B = 11 are set for lack of further digits):

2, 3, 5, 7, B, 11, 15, 57, 5B, 111, 117, 11B, 555B, R ₅ , R ₁₇ , R ₈₁ , R ₉₁ , R ₂₂₅ , R ₂₅₅ , R _4A5 , ...

There are no other permutable prime numbers with base 12 with fewer than 9739 digits. There are also no n -digit permutable prime numbers with base 12 with 4 <n <12 ¹⁴⁴ , which is not a repunit.

• The smallest permutable prime numbers with base b with more than two different digits are the following (where A = 10, B = 11, ..., E = 14, ... is set for lack of further digits):

139 ₁₁ , 36A ₁₁ , 247 ₁₃ , 78A ₁₃ , 29E ₁₉

There are no other numbers of this form that are smaller than 10 ⁹ (M. Fiorentini, 2015).

properties

• Permutable prime numbers with base 2 must be repunits , i.e. they can only contain ones.

(If there were a zero there, one could give the zero to the ones place by permutating their places and the number would be even and not prime.)

• Permutable prime numbers with base 2 are the Mersenne numbers .
• Single-digit prime numbers, regardless of the base, are permutable prime numbers for trivial reasons.
• Permutable primes with more than one place, regardless of the base, only allowed digits possess the prime are the base.

(In the decimal system, i.e. with base 10, the digits 0, 2, 4, 5, 6 and 8 must not appear because they are not prime to 10.)

• Permutable prime numbers with base 10 or base 12 are either repunits or near-repunits. If this prime number has n places, either the same digit is always found in all n places, or the same digit is used in n-1 places and another digit is only in one place (for example xxxxxy ). It therefore consists of a maximum of two different digits.
• Permutable prime numbers with base 10 or base 12 only have digits that are relatively prime.

(If there were a prime p sharing both different digits, that prime would divide the whole number.)

• Permutable prime numbers with base 10 or base 12, in which all digits are the same (i.e. repunits), always consist of ones.
• Every permutable prime number is also a circular prime number at the same time .
• Not every circular prime is a permutable prime.

Unsolved problems

• Are there other permutable prime numbers with base 12 that are not repunits and that are not in the following list:

2, 3, 5, 7, B, 15, 57, 5B, 117, 11B, 555B

It is believed that there are no more.

See also

• Special prime numbers and prime number constellations
• List of special numbers
• Circular prime number

Individual evidence

1. ↑ ^{a b} H.E. Richert: Om permutable primary. In: Norsk Matematisk Tidsskrift. No. 33, 1951, p. 50 ff.
2. ↑ T. Bhargava, P. Doyle: On the existence of absolute primes. In: Mathematics Magazine. No. 47, 1974, p. 233.
3. ↑ Chris K. Caldwell: The Prime Glossery: Permutable Prime . Retrieved August 11, 2020.
4. ^ OEIS: Absolute primes, alternative definition . Retrieved February 24, 2014.
5. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, Search for: (10 ^ x-1) / 9. PRP Records, accessed July 8, 2018 . 
6. ↑ A. Slinko: primes Absolute . Retrieved February 24, 2014.
7. ↑ AW Johnson: Absolute primes. In: Mathematics Magazine. No. 50, 1977, p. 100 ff.