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- Eine permutierbare Primzahl (auch absolute Primzahl) ist eine Primzahl, bei der eine beliebige Neuanordnung ihrer Ziffern ebenfalls eine Primzahl ergibt. Zum Beispiel ist 113 eine permutierbare Primzahl, da 131 und 311 ebenfalls prim sind. Ob diese Bedingung erfüllt ist, hängt dabei auch vom verwendeten Stellenwertsystem ab. Als sich erstmals der Mathematiker Hans-Egon Richert in einem Aufsatz mit diesen Zahlen befasste, nannte er sie permutierbare Primzahlen. Spätere Autoren verwendeten auch den Begriff der absoluten Primzahl. (de)
- Un número primo permutable, también conocido como primo anagramático, es un número primo que, en una base dada, puede cambiar las posiciones de sus dígitos a través de cualquier permutación y seguir siendo un número primo. H. E. Richert, quien posiblemente fue el primero en estudiar estos primos, los llamó primos permutables, pero luego también se les llamó primos absolutos. (es)
- A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes. In base 10, all the permutable primes with fewer than 49,081 digits are known 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... (sequence in the OEIS) Of the above, there are 16 unique permutation sets, with smallest elements 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ... (sequence in the OEIS) Note Rn = is a repunit, a number consisting only of n ones (in base 10). 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 proven 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 conjectured that there are no non-repunit permutable primes other than the eighteen listed above. They can be split into seven permutation sets: {13, 31}, {17, 71}, {37, 73}, {79, 97}, {113, 131, 311}, {199, 919, 991}, {337, 373, 733}. In base 2, only repunits can be permutable primes, because any 0 permuted to the ones place results in an even number. Therefore, the base 2 permutable primes are the Mersenne primes. 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 trivially permutable. In base 12, the smallest elements of the unique permutation sets of the permutable primes with fewer than 9,739 digits are known (using inverted two and three for ten and eleven, respectively) 2, 3, 5, 7, Ɛ, R2, 15, 57, 5Ɛ, R3, 117, 11Ɛ, 555Ɛ, R5, R17, R81, R91, R225, R255, R4ᘔ5, ... There is no n-digit permutable prime in base 12 for 4 < n < 12144 which is not a repunit. It is conjectured that there are no non-repunit permutable primes in base 12 other than those listed above. In base 10 and base 12, every permutable prime is a repunit or a near-repdigit, that is, it is a permutation of the integer P(b, n, x, y) = xxxx...xxxyb (n digits, in base b)where x and y are digits which is coprime to b. Besides, x and y must be also coprime (since if there is a prime p divides both x and y, then p also divides the number), so if x = y, then x = y = 1. (This is not true in all bases, but exceptions are rare and could be finite in any given base; the only exceptions below 109 in bases up to 20 are: 13911, 36A11, 24713, 78A13, 29E19 (M. Fiorentini, 2015).) Let P(b, n, x, y) be a permutable prime in base b and let p be a prime such that n ≥ p. If b is a primitive root of p, and p does not divide x or |x - y|, then n is a multiple of p - 1. (Since b is a primitive root mod p and p does not divide |x − y|, the p numbers xxxx...xxxy, xxxx...xxyx, xxxx...xyxx, ..., xxxx...xyxx...xxxx (only the bp−2 digit is y, others are all x), xxxx...yxxx...xxxx (only the bp−1 digit is y, others are all x), xxxx...xxxx (the repdigit with n xs) mod p are all different. That is, one is 0, another is 1, another is 2, ..., the other is p − 1. Thus, since the first p − 1 numbers are all primes, the last number (the repdigit with n xs) must be divisible by p. Since p does not divide x, so p must divide the repunit with n 1s. Since b is a primitive root mod p, the multiplicative order of n mod p is p − 1. Thus, n must be divisible by p − 1) Thus, if b = 10, the digits coprime to 10 are {1, 3, 7, 9}. Since 10 is a primitive root mod 7, so if n ≥ 7, then either 7 divides x (in this case, x = 7, since x ∈ {1, 3, 7, 9}) or |x − y| (in this case, x = y = 1, since x, y ∈ {1, 3, 7, 9}. That is, the prime is a repunit) or n is a multiple of 7 − 1 = 6. Similarly, since 10 is a primitive root mod 17, so if n ≥ 17, then either 17 divides x (not possible, since x ∈ {1, 3, 7, 9}) or |x − y| (in this case, x = y = 1, since x, y ∈ {1, 3, 7, 9}. That is, the prime is a repunit) or n is a multiple of 17 − 1 = 16. Besides, 10 is also a primitive root mod 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, ..., so n ≥ 17 is very impossible (since for this primes p, if n ≥ p, then n is divisible by p − 1), and if 7 ≤ n < 17, then x = 7, or n is divisible by 6 (the only possible n is 12). If b = 12, the digits coprime to 12 are {1, 5, 7, 11}. Since 12 is a primitive root mod 5, so if n ≥ 5, then either 5 divides x (in this case, x = 5, since x ∈ {1, 5, 7, 11}) or |x − y| (in this case, either x = y = 1 (That is, the prime is a repunit) or x = 1, y = 11 or x = 11, y = 1, since x, y ∈ {1, 5, 7, 11}.) or n is a multiple of 5 − 1 = 4. Similarly, since 12 is a primitive root mod 7, so if n ≥ 7, then either 7 divides x (in this case, x = 7, since x ∈ {1, 5, 7, 11}) or |x − y| (in this case, x = y = 1, since x, y ∈ {1, 5, 7, 11}. That is, the prime is a repunit) or n is a multiple of 7 − 1 = 6. Similarly, since 12 is a primitive root mod 17, so if n ≥ 17, then either 17 divides x (not possible, since x ∈ {1, 5, 7, 11}) or |x − y| (in this case, x = y = 1, since x, y ∈ {1, 5, 7, 11}. That is, the prime is a repunit) or n is a multiple of 17 − 1 = 16. Besides, 12 is also a primitive root mod 31, 41, 43, 53, 67, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, ..., so n ≥ 17 is very impossible (since for this primes p, if n ≥ p, then n is divisible by p − 1), and if 7 ≤ n < 17, then x = 7 (in this case, since 5 does not divide x or x − y, so n must be divisible by 4) or n is divisible by 6 (the only possible n is 12). (en)
- En arithmétique, un nombre premier permutable est un nombre premier qui, dans une base donnée, reste premier après n'importe quelle permutation de ses chiffres. Cette définition a été proposée par Bill Gosper en 2003. (fr)
- 置換可能素数（ちかんかのうそすう、英語: permutable prime）は、与えられたにおいて、任意の桁の数字を置換しても素数となる素数のことである。この素数を最初に研究したはこれを"permutable primes"（置換可能素数）と呼んだが、後に"absolute primes"（絶対素数）とも呼ばれた。また、"anagrammatic prime"（アナグラム素数）とも呼ばれる。 基数10においては、49,081桁以下の全ての置換可能素数が判明している。 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... オンライン整数列大辞典の数列 A003459 上記から、置換により同じ数字となるもののうち最小のもの以外を除くと、以下の16個となる。 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ... オンライン整数列大辞典の数列 A258706 ここで、 Rn = は、n個の1（基数10）だけで構成される数（レピュニット数）である。全てのレピュニット素数は上記に定義した置換可能素数であるが、定義によっては少なくとも2つの異なる桁が必要となる。 全ての2桁以上の置換可能素数は1,3,7,9で構成されている。これは、2以外の偶数は素数ではなく、5以外の素数は5で割り切れないからである。1,3,7,9の4つの数字のうちの3つを含む置換可能素数が存在しないこと、1,3,7,9から選択された2つの数字の各々が2つ以上から構成される置換可能素数が存在しないことは証明されている。 3 < n < 6·10175となるn桁のレピュニット以外の置換可能素数は存在しない。上記に挙げた以外のレピュニットでない置換可能素数は存在しないと予想されている。 (ja)
- 재배열 가능 소수 또는 치환 가능 소수는 주어진 진법에서, 그 자릿수를 가능한 여러 가지 순열로 바꾸어도 여전히 소수인 소수 (수론)를 말한다. 이 소수를 처음 연구한 것으로 알려진 는 재배열 가능 소수(또는 순열 소수)라고 이름 지었으나, 나중에는 절대 소수라고도 불렀다.. 또한 자릿수에 2, 4, 6, 8이 있는 소수는 자릿수를 재배열 하면 짝수이고, 5가 있으면 5의 배수가 되므로 재배열 가능 소수가 아니다. 또한 n진법에서 n보다 작아서 한 자리인 소수는 배열하는 방법이 한가 지이 므로 무조건 재배열 가능 소수에 속하며, n진법에서 자리수가 모두 1로 되어있는 단위 반복 소수 역시 배열하는 반법이 한 가지 밖에 없으므로 무조건 재배열 가능 소수가 된다. 10진법에서, 49,081 자릿수 이하의 자릿수에서의 모든 재배열 소수들은 다음과 같다. 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, 1111111111111111111 (R19), 11111111111111111111111 (R23), R317, R1031, R49081 , ... (OEIS의 수열 ) 자릿수를 재배열해서 나오는 소수들을 같은 것으로 보면 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, , R23, R317, R1031 이렇게 되어 337 다음에 재배열 가능 소수가 한참동안 나오지 않았다가 1이 19개 늘어서있는 이 나오고, 그 뒤로는 1이 23개 늘어선 이 나오는 식이다. 여기서 Rn = 이고, 단위 반복 소수의 일종이다. (ko)
- Un primo permutabile è un numero primo tale che, in una data base di numerazione, qualunque permutazione delle sue cifre formi ancora un numero primo. In base 10, la sequenza dei primi permutabili inizia come segue (sequenza nell'OEIS): 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19, R23 Dove Ri indica un repunit di i cifre. Ogni repunit primo è evidentemente un primo permutabile. In base 2, solo i repunit possono essere primi permutabili, perché ogni 0 spostato in ultima posizione genererebbe un numero pari. Ciò si può generalizzare ad ogni sistema di numerazione in base pari (come quello decimale o esadecimale): tutti i primi permutabili (tranne 2) contengono solamente cifre dispari. (it)
- 可交換質數（permutable prime）是指一個質數，在特定進制下的各位數字可以任意交換位置，其結果仍為質數。數學家Hans-Egon Richert最早研究這類的質數，命名為可交換質數，不過這類質數也被稱為絕對質數（absolute primes）。 以下是十進制下所有已知的，小於49081位數的可交換質數（OEIS數列）： 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031。 以上有些質數的的數字相同，只是位置不同，例如13和31，若這類由同一質數交換位置所得的質數只用一個作為代表，那麼只有16組可交換質數： 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031. 其中Rn = 是循環單位，是由n個1組成的（十進位）數字。循環單位的質數是可交換質數，不過也有些可交換質數的定義中包括至少有二個不同的數字，此定義下循環單位的質數就不是可交換質數。 所有超過1位數的可交換質數都是由1,3,7,9數字組成，不包括所有偶數及5，因為若有出現這些數字，這些數字在交換位置後可能會在個位數，而超過1位數的數字，若個位數為偶數或是5，一定不是質數。已有數字家證明沒有任一個可交換質數中有1,3,7,9中的三個數字，也沒有任一個可交換質數其中有1,3,7,9中的二個數字，且每個數字出現不止一次。 對於3 < n < 6·10175的正整數n，不存在n位數且不是循環單位的可交換質數。目前猜想除了上述數字外，不存在其他的可交換質數。 在二進制中，只有循環單位才可能是可交換質數，因此若任何一位數為0，這個0交換位置到最末位時，數字是合數，不是質數。因此二進制的可交換質數即為梅森素数。此概念可以延伸到其他進位制中，一位數的質數必定是可交換質數，而超過一位數可交換質數的各位數字一定是由和進位制基數互質的數所組成。 (zh)
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- Eine permutierbare Primzahl (auch absolute Primzahl) ist eine Primzahl, bei der eine beliebige Neuanordnung ihrer Ziffern ebenfalls eine Primzahl ergibt. Zum Beispiel ist 113 eine permutierbare Primzahl, da 131 und 311 ebenfalls prim sind. Ob diese Bedingung erfüllt ist, hängt dabei auch vom verwendeten Stellenwertsystem ab. Als sich erstmals der Mathematiker Hans-Egon Richert in einem Aufsatz mit diesen Zahlen befasste, nannte er sie permutierbare Primzahlen. Spätere Autoren verwendeten auch den Begriff der absoluten Primzahl. (de)
- Un número primo permutable, también conocido como primo anagramático, es un número primo que, en una base dada, puede cambiar las posiciones de sus dígitos a través de cualquier permutación y seguir siendo un número primo. H. E. Richert, quien posiblemente fue el primero en estudiar estos primos, los llamó primos permutables, pero luego también se les llamó primos absolutos. (es)
- En arithmétique, un nombre premier permutable est un nombre premier qui, dans une base donnée, reste premier après n'importe quelle permutation de ses chiffres. Cette définition a été proposée par Bill Gosper en 2003. (fr)
- A permutable prime, also known as anagrammatic prime, is a prime number which, in a given base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to study these primes, called them permutable primes, but later they were also called absolute primes. In base 10, all the permutable primes with fewer than 49,081 digits are known 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... (sequence in the OEIS) (en)
- Un primo permutabile è un numero primo tale che, in una data base di numerazione, qualunque permutazione delle sue cifre formi ancora un numero primo. In base 10, la sequenza dei primi permutabili inizia come segue (sequenza nell'OEIS): 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19, R23 Dove Ri indica un repunit di i cifre. Ogni repunit primo è evidentemente un primo permutabile. (it)
- 재배열 가능 소수 또는 치환 가능 소수는 주어진 진법에서, 그 자릿수를 가능한 여러 가지 순열로 바꾸어도 여전히 소수인 소수 (수론)를 말한다. 이 소수를 처음 연구한 것으로 알려진 는 재배열 가능 소수(또는 순열 소수)라고 이름 지었으나, 나중에는 절대 소수라고도 불렀다.. 또한 자릿수에 2, 4, 6, 8이 있는 소수는 자릿수를 재배열 하면 짝수이고, 5가 있으면 5의 배수가 되므로 재배열 가능 소수가 아니다. 또한 n진법에서 n보다 작아서 한 자리인 소수는 배열하는 방법이 한가 지이 므로 무조건 재배열 가능 소수에 속하며, n진법에서 자리수가 모두 1로 되어있는 단위 반복 소수 역시 배열하는 반법이 한 가지 밖에 없으므로 무조건 재배열 가능 소수가 된다. 10진법에서, 49,081 자릿수 이하의 자릿수에서의 모든 재배열 소수들은 다음과 같다. 여기서 Rn = 이고, 단위 반복 소수의 일종이다. (ko)
- 置換可能素数（ちかんかのうそすう、英語: permutable prime）は、与えられたにおいて、任意の桁の数字を置換しても素数となる素数のことである。この素数を最初に研究したはこれを"permutable primes"（置換可能素数）と呼んだが、後に"absolute primes"（絶対素数）とも呼ばれた。また、"anagrammatic prime"（アナグラム素数）とも呼ばれる。 基数10においては、49,081桁以下の全ての置換可能素数が判明している。 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031, ... オンライン整数列大辞典の数列 A003459 上記から、置換により同じ数字となるもののうち最小のもの以外を除くと、以下の16個となる。 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031, ... オンライン整数列大辞典の数列 A258706 (ja)
- 可交換質數（permutable prime）是指一個質數，在特定進制下的各位數字可以任意交換位置，其結果仍為質數。數學家Hans-Egon Richert最早研究這類的質數，命名為可交換質數，不過這類質數也被稱為絕對質數（absolute primes）。 以下是十進制下所有已知的，小於49081位數的可交換質數（OEIS數列）： 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, R19 (1111111111111111111), R23, R317, R1031。 以上有些質數的的數字相同，只是位置不同，例如13和31，若這類由同一質數交換位置所得的質數只用一個作為代表，那麼只有16組可交換質數： 2, 3, 5, 7, R2, 13, 17, 37, 79, 113, 199, 337, R19, R23, R317, R1031. 其中Rn = 是循環單位，是由n個1組成的（十進位）數字。循環單位的質數是可交換質數，不過也有些可交換質數的定義中包括至少有二個不同的數字，此定義下循環單位的質數就不是可交換質數。 對於3 < n < 6·10175的正整數n，不存在n位數且不是循環單位的可交換質數。目前猜想除了上述數字外，不存在其他的可交換質數。 (zh)
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