Circular prime number

In number theory , a circular prime number (from circular prime ) is a prime number whose digits can be exchanged cyclically and the number obtained remains a prime number.

Examples in the decimal system

• In the decimal system, the number is a circular prime number, because the following prime numbers are obtained by cyclically interchanging their digits: ${\displaystyle p=1193}$

${\displaystyle p=1193\in \mathbb {P} \longrightarrow p_{2}=1931\in \mathbb {P} \longrightarrow p_{3}=9311\in \mathbb {P} \longrightarrow p_{4}=3119\in \mathbb {P} \longrightarrow p}$

• The following circular prime numbers are known in the decimal system (from which you can make more by cyclic exchange):

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, R ₁₉ = 1111111111111111111, R ₂₃ = 11111111111111111111111, R ₃₁₇ , R ₁₀₃₁ , R ₄₉₀₈₁ , R ₈₆₄₅₃ , R ₁₀₉₂₉₇ and R ₂₇₀₃₄₃ (series A016114 in OEIS )

There is a total of ones (it has places). These numbers are called repunits . 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. There are no other circular prime numbers that are less than . ${\displaystyle R_{n}=11\ldots 11}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle R_{49081},R_{86453},R_{109297}}$ ${\displaystyle R_{270343}}$ ${\displaystyle 10^{23}}$

• The following circular prime numbers are known in the decimal system (including those obtained through cyclic interchange, but without the larger repunits):

2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939, 393919, 919393, 933199, 939193, 939391, 993319, 999331 (sequence A068652 in OEIS )

Apart from the repunits, there are probably no other circular prime numbers.

Examples in other bases

• The following circular prime numbers are known in the duodecimal system (i.e. with a base ) (which can be made into more by cyclic exchange). In the absence of further digits, A = 10 and B = 11 are set: ${\displaystyle b=12}$

2, 3, 5, 7, B, 11, 15, 57, 5B, 111, 117, 11B, 175, 1B7, 157B, 555B, 11111, 115B77, R ₁₇ , R ₈₁ , R ₉₁ , R ₂₂₅ , R ₂₅₅ , R _4A5 , R ₅₇₇₇ , R _879B , R _198B1 , R ₂₃₁₇₅ and R ₃₁₁₄₀₇

As before, there is a repunit (for the base ) with a total of ones (it has places). There are no more circular prime numbers in the duodecimal system. ${\displaystyle R_{n}=11\ldots 11}$ ${\displaystyle b=12}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle 12^{12}}$

Example:

It is the duodecimal number a prime number. If you swap their digits cyclically, you get three more prime numbers, namely , and . ${\displaystyle 157B_{12}={\underline {1}}\cdot 12^{3}+{\underline {5}}\cdot 12^{2}+{\underline {7}}\cdot 12^{1}+{\underline {11}}\cdot 12^{0}=2543\in \mathbb {P} }$ ${\displaystyle 57B1_{12}=9781\in \mathbb {P} }$ ${\displaystyle 7B15_{12}=13697\in \mathbb {P} }$ ${\displaystyle B157_{12}=19219\in \mathbb {P} }$

properties

• In the decimal system, except for and, a circular prime number may only consist of the digits or . ${\displaystyle p=2}$ ${\displaystyle p=5}$ ${\displaystyle 1,3,7}$ ${\displaystyle 9}$

Proof:

If the digits or were also allowed for a circular prime number , they could be swapped cyclically until these digits are in the ones place. But then they would be divisible by or and thus no longer prime numbers. ${\displaystyle 0,2,4,5,6}$ ${\displaystyle 8}$ ${\displaystyle p=2}$ ${\displaystyle p=5}$ ${\displaystyle \Box }$

• Every prime repunit is a circular prime.

Proof:

A repunit consists entirely of ones (such as ). Thus, by interchanging their digits cyclically, no other number is obtained, so the "newly received" number remains a prime number. ${\displaystyle R_{19}=1111111111111111111}$ ${\displaystyle \Box }$

• Let be a circular prime number in the dual system (i.e. with basis ). Then: ${\displaystyle n}$ ${\displaystyle b=2}$

${\displaystyle n}$is a Mersenne prime number .

Proof:

In the dual system there are only the two digits and . If one occurs in a binary number, one would achieve by cyclic exchange that it is in the ones place. Dual numbers with one in the ones place are even numbers and therefore not prime numbers (for example is an even number). So a circular prime number can not contain any zeros as a base , so it must consist exclusively of ones. However, binary numbers that only consist of ones, converted into the decimal system, have the form (for example is ). These numbers are Mersenne numbers. If they're prime, they are Mersenne primes. ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 11110_{2}={\underline {1}}\cdot 2^{4}+{\underline {1}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {0}}\cdot 2^{0}=16+8+4+2+0=30}$ ${\displaystyle b=2}$ ${\displaystyle n=2^{n}-1}$ ${\displaystyle 11111_{2}={\underline {1}}\cdot 2^{4}+{\underline {1}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {1}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=16+8+4+2+1=31=2^{5}-1}$ ${\displaystyle \Box }$

• Every permutable prime number is a circular prime number.

Proof:

A permutable prime number is a prime number in which you can rearrange its digits as you like and still get a prime number. In the case of circular prime numbers, however, only a cyclic exchange is required, which is of course also permitted for permutable prime numbers (cyclic exchanges are just a slightly more special rearrangement of their digits). Thus a permutable prime number is always a circular number. ${\displaystyle \Box }$

• Not every circular prime is a permutable prime.

Proof:

A counterexample is sufficient: If the digits of the circular prime number are swapped appropriately, the composite number is obtained . Thus is not a permutable prime number. ${\displaystyle p=1193}$ ${\displaystyle 1139=17\cdot 67}$ ${\displaystyle p=1193}$ ${\displaystyle \Box }$

Unsolved problems

• It is assumed that there are infinitely many circular prime numbers because there are probably infinitely many prime repunits, all of which are circular prime numbers at the same time.

• It is assumed that there are no further circular prime numbers that are not repunits at the same time.

Individual evidence

1. ↑ Eric W. Weisstein : Circular Prime . In: MathWorld (English).
2. ^ Henri Lifchitz, Renaud Lifchitz: PRP Records - Probable Primes Top 10000, Search for: (10 ^ x-1) / 9. PRP Records, accessed July 8, 2018 . 
3. ^ Patrick De Geest: Circular Primes. World Of Numbers, accessed July 8, 2018 . 
4. ↑ ^{a b c} Chris K. Caldwell: Circular Prime. Prime Pages, accessed July 8, 2018 . 

Web links

• Eric W. Weisstein : Circular Prime . In: MathWorld (English).
• Chris K. Caldwell: Circular Prime. Prime Pages, accessed July 8, 2018 .