A **permutable prime**, also known as **anagrammatic prime**, is a prime numberwhich, 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**.

All permutable primes of two or more digits are composed from the digits 1, 5, 7, E, because no prime number except 2 is even, and no prime number besides 3 is divisible by 3. It is proven that no permutable prime exists which contains three different of the four digits 1, 5, 7, E, as well as that there exists no permutable prime composed of two or more of each of two digits selected from 1, 5, 7, E.

The smallest elements of the unique permutation sets of the permutable primes with fewer than 5777 digits are known

- 2, 3, 5, 7, E, R
_{2}, 15, 57, 5E, R_{3}, 117, 11E, 555E, R_{5}, R_{17}, R_{81}, R_{91}, R_{225}, R_{255}, R_{4X5}

where R_{n} = (10^n-1)/E is the repunit with length n, e.g. R_{3} = 111.

There is no *n*-digit permutable prime for 4 < *n* < 10^{100} which is not a repunit. It is conjectured that there are no non-repunit permutable primes other than those listed above.

In base 10, 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*...*xxxy _{b}* (

*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 in bases up to 20 are 139

_{E}, 36X

_{E}, 2:4:7

_{11}, 7:8:X

_{11}, 2:9:12

_{17}, 1:5:E

_{19}, 2:8:11

_{19}, 4:9:16

_{1E}, and 6:12:13

_{1E}, and there are no exceptions that is not 3-digit numbers in all bases up to 36, and it is also conjectured that this is also true in all bases, i.e. in any base, all exceptions are 3-digit numbers)

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 *b ^{p}*

^{−2}digit is

*y*, others are all

*x*),

*xxxx*...

*yxxx*...

*xxxx*(only the

*b*

^{p}^{−1}digit is

*y*, others are all

*x*),

*xxxx*...

*xxxx*(the repdigit with

*n*

*x*s) 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

*nx*s) 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, in the dozenal (base 10) case, if a non-repunit permutable prime has >= 15 digits, then the number of digits must be a multiple of 14 (since 15 cannot divide either *x* or |x−*y*| (since *x* ∈ {1, 5, 7, E} and *y* ∈ {1, 5, 7, E}) and 10 is a primitive root mod 15), thus the number of digits must be at least 28 (=2*14), however, if a non-repunit permutable prime has >= 27 digits, then the number of digits must be a multiple of 26, thus the number of digits must be at least lcm(14,26) = 180, ..., and the possibility getting rare and rare.

base | conjectured largest non-repunit permutable prime | dozenal value |
---|---|---|

3 | 21 | 7 |

4 | 311 | 45 |

5 | 44441 | 1,981 |

6 | 551 | 157 |

7 | 5554 | 1,1X7 |

8 | 7333 | 2,24E |

9 | 8777 | 3,8E1 |

X | 991 | 6X7 |

E | XXXX7 | 79,247 |

10 | E555 | E,555 |

11 | 7:6:6:6 | 9,651 |

12 | 9:1:1:1 | 12,4E7 |

13 | 11:7:7:7:7 | 28E,611 |

14 | 11:11:E | 2,077 |

(there are no non-repunit permutable prime for base < 3)

base | non-repunit permutable prime | dozenal value |
---|---|---|

2 | ||

3 | 2, 12, 21 | 2, 5, 7 |

4 | 2, 3, 13, 31, 113, 131, 311 | 2, 3, 7, 11, 1E, 25, 45 |

5 | 2, 3, 12, 21, 23, 32, 34, 43, 14444, 41444, 44144, 44414, 44441 | 2, 3, 7, E, 11, 15, 17, 1E, 881, 1711, 1921, 1971, 1981 |

6 | 2, 3, 5, 15, 51, 155, 515, 551 | 2, 3, 5, E, 27, 5E, 13E, 157 |

7 | 2, 3, 5, 14, 16, 23, 25, 32, 41, 52, 56, 61, 65, 155, 166, 515, 551, 616, 661, 1444, 4144, 4414, 4441, 4555, 5455, 5545, 5554 | 2, 3, 5, E, 11, 15, 17, 1E, 25, 31, 35, 37, 3E, 75, 81, 195, 1E5, 217, 241, 3E7, X11, XE7, E11, E61, 1167, 11X1, 11X7 |

8 | 2, 3, 5, 7, 15, 35, 37, 51, 53, 57, 73, 75, 3337, 3373, 3733, 7333 | 2, 3, 5, 7, 11, 25, 27, 35, 37, 3E, 4E, 51, 1027, 104E, 11E7, 224E |

9 | 2, 3, 5, 7, 12, 14, 18, 21, 25, 41, 47, 52, 74, 78, 81, 87, 122, 212, 221, 788, 878, 887, 4555, 5455, 5545, 5554, 7778, 7787, 7877, 8777 | 2, 3, 5, 7, E, 11, 15, 17, 1E, 31, 37, 3E, 57, 5E, 61, 67, 85, 125, 131, 45E, 4EE, 507, 1E4E, 23XE, 244E, 2457, 33X5, 33E1, 3451, 38E1 |

X | 2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991 | 2, 3, 5, 7, 11, 15, 27, 31, 5E, 61, 67, 81, 95, XE, 147, 21E, 241, 271, 511, 647, 6X7 |

E | 2, 3, 5, 7, 12, 16, 18, 21, 27, 29, 34, 3X, 43, 49, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, X3, 117, 139, 171, 193, 1XX, 319, 335, 353, 36X, 391, 3X6, 533, 566, 588, 63X, 656, 665, 6X3, 711, 7XX, 858, 885, 913, 931, X1X, X36, X63, X7X, XX1, XX7, 2777, 7277, 7727, 7772, 9XXX, X9XX, XX9X, XXX9, 7XXXX, X7XXX, XX7XX, XXX7X, XXXX7 | 2, 3, 5, 7, 11, 15, 17, 1E, 25, 27, 31, 37, 3E, 45, 51, 57, 5E, 61, 67, 6E, 75, 81, 85, 87, 8E, 95, E7, 117, 147, 167, 181, 27E, 295, 2E1, 307, 327, 33E, 455, 485, 4X5, 541, 557, 565, 59E, 5E7, 687, 71E, 745, 77E, 797, 867, 881, 8X7, 901, 921, 927, 20E5, 56E7, 5X95, 5E17, 7851, 849E, 8571, 857E, 57947, 76X81, 78EX7, 79221, 79247 |

10 | 2, 3, 5, 7, E, 15, 51, 57, 5E, 75, E5, 117, 11E, 171, 1E1, 711, E11, 555E, 55E5, 5E55, E555 | 2, 3, 5, 7, E, 15, 51, 57, 5E, 75, E5, 117, 11E, 171, 1E1, 711, E11, 555E, 55E5, 5E55, E555 |

11 | 2, 3, 5, 7, E, 1:4, 1:6, 1:X, 2:3, 2:5, 3:2, 3:8, 4:1, 5:2, 5:6, 5:8, 6:1, 6:5, 6:E, 7:X, 7:10, 8:3, 8:5, 9:X, X:1, X:7, X:9, E:6, 10:7, 1:1:E, 1:3:3, 1:5:5, 1:E:1, 2:2:9, 2:4:7, 2:7:4, 2:9:2, 3:1:3, 3:3:1, 3:3:E, 3:8:8, 3:E:3, 4:2:7, 4:7:2, 5:1:5, 5:5:1, 7:2:4, 7:4:2, 7:7:9, 7:8:X, 7:9:7, 7:X:8, 8:3:8, 8:7:X, 8:8:3, 8:X:7, 9:2:2, 9:7:7, X:7:8, X:8:7, E:1:1, E:3:3, 4:4:4:5, 4:4:5:4, 4:5:4:4, 5:4:4:4, 6:6:6:7, 6:6:7:6, 6:7:6:6, 7:6:6:6 | 2, 3, 5, 7, E, 15, 17, 1E, 25, 27, 35, 3E, 45, 57, 5E, 61, 67, 6E, 75, 85, 87, 8E, 91, X7, XE, E5, E7, 105, 117, 141, 157, 17E, 221, 271, 291, 301, 321, 377, 397, 3X5, 437, 465, 4E1, 541, 5EE, 63E, 851, 871, 8XE, 901, 90E, 921, 987, X11, X17, X41, X91, E2E, 1051, 1061, 1101, 1125, 5615, 5625, 5735, 6945, 8321, 8331, 8441, 9651 |

12 | 2, 3, 5, 7, E, 11, 1:3, 1:5, 1:9, 3:1, 3:5, 3:E, 5:1, 5:3, 5:9, 9:1, 9:5, 9:E, 9:11, E:3, E:9, E:11, 11:9, 11:E, 3:3:11, 3:11:3, 11:3:3, 1:1:1:9, 1:1:9:1, 1:9:1:1, 9:1:1:1 | 2, 3, 5, 7, E, 11, 15, 17, 1E, 37, 3E, 45, 5E, 61, 67, X7, XE, E5, E7, 111, 117, 11E, 13E, 141, 457, 545, 1601, 186E, 1937, 274E, 124E7 |

13 | 2, 3, 5, 7, E, 11, 1:2, 1:4, 1:12, 2:1, 2:7, 2:E, 2:11, 4:1, 4:7, 4:11, 7:2, 7:4, 7:8, 8:7, 8:E, E:2, E:8, 11:2, 11:4, 12:1, 1:11:11, 2:2:7, 2:7:2, 4:4:E, 4:E:4, 7:2:2, 7:7:11, 7:11:7, E:4:4, E:12:12, 11:1:11, 11:7:7, 11:11:1, 12:E:12, 12:12:E, 1:4:4:4, 4:1:4:4, 4:4:1:4, 4:4:4:1, 7:7:7:7:11, 7:7:7:11:7, 7:7:11:7:7, 7:11:7:7:7, 11:7:7:7:7 | 2, 3, 5, 7, E, 11, 15, 17, 25, 27, 31, 35, 37, 51, 57, 61, 8E, 91, 95, X7, XE, 11E, 125, 145, 147, 157, 301, 347, 3X5, 68E, 751, E1E, E91, 1041, 1577, 168E, 1861, 1911, 1981, 1E15, 1E4E, 2617, 7E91, 8417, 8451, 163891, 163941, 164611, 173541, 28E611 |

14 | 2, 3, 5, 7, E, 11, 1:2, 1:4, 1:12, 2:1, 2:7, 2:E, 2:11, 4:1, 4:7, 4:11, 7:2, 7:4, 7:8, 8:7, 8:E, E:2, E:8, 11:2, 11:4, 12:1, 1:11:11, 2:2:7, 2:7:2, 4:4:E, 4:E:4, 7:2:2, 7:7:11, 7:11:7, E:4:4, E:12:12, 11:1:11, 11:7:7, 11:11:1, 12:E:12, 12:12:E, 1:4:4:4, 4:1:4:4, 4:4:1:4, 4:4:4:1, 7:7:7:7:11, 7:7:7:11:7, 7:7:11:7:7, 7:11:7:7:7, 11:7:7:7:7 | 2, 3, 5, 7, E, 11, 1E, 27, 45, 4E, 51, 6E, 75, 95, 105, 12E, 13E, 157, 181, 18E, 1E1, 1E7, 241, 301, 61E, 901, 965, X45, 109E, 112E, 118E, 1781, 184E, 189E, 1911, 1E85, 2051, 2077 |