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Introduction

Circular Primes are primes with a special property.
One can find them by repeatedly chopping away the leftmost digit and appending them to the other end of the number.
Repeat this process until you come back at the starting number.
If all the intermediate formed numbers are prime then the starting number can be called 'circular'.
An example (e.g. with the number 1193) makes everthing clear
 
1193 is prime
1931 is prime
9311 is prime
3119 is prime
1193 back to starting position

Circular Primes

It is immediate clear that a circular prime only consists of the digits 1, 3, 7 and 9.
The single-digit primes 2 and 5 are the only exceptions.
In the rightmost column of the table I put the total number of primes of the given length made of a mixture
of these four digits. Only these combinations are eligible to become a circular prime however small the chances may be.
An even digit arriving at the end of the number obviously makes the number composite.
Ending with 5 evidently makes the number divisible by 5.
Also mentioned, in case no circular primes were found, are the -near misses- meaning that only one combination failed.
More near misses exists when the digits 2, 4, 5, 6 and 8 are allowed. But that falls outside the scope of this webpage (A270083).
( Carlos Rivera examined our case (only digits 1, 3, 7 and 9) in more detail : see Sloane's A045978 ).

Statistically speaking, it seems very unlikely that I will discover more of them as the length increases beyond 14.
Should one exist above length 14, the chances are very high that this number will become famous in numbertheory circles.
It might be a good idea to take Keith Devlin words into consideration.
I quote from his book All the Math that's Fit to Print, chapter 97 :

"For example, there are the so-called permutable primes. These are prime numbers that remain prime
when you rearrange their digits in any order you please. For example, 13 is a permutable prime, since
both 13 and 31 are prime. Again, 113 is a permutable prime, since it and each of the numbers 131 and
311 is prime. It is known that there are only seven such numbers within reasonable range (less than about
4 followed by 467 zeros, in fact). You now know two of them. Find the other five."

John Shonder (email) informed me on [ August 18, 1997 ] that the  repunit primes  are also circular primes by the definition
given above. Every single "permutation" of those repunital digits makes a prime, meaning that any arrangement of their digits
results in a prime. He admits that they are trivial or 'obvious' circular but they qualify just as well.
Indeed he's right and therefore I included them in the next table. Up to this moment only five repunit primes are known :
R2, R19, R23, R317 and R1031 (See Sloane's Encyclopedia of Integer Sequences).
The last one was discovered in 1986 by Williams and Dubner.
Furthermore these repunit primes are palindromic so there you have a connection with the rest of my website.
Hence I can talk about Circular Repunital Palindromic Primes.

Justin Chan's (email)
contribution
[ July 11, 2008 ]

I found this article at http://primes.utm.edu/ :

http://www.lacim.uqam.ca/~plouffe/OEIS/archive_in_pdf/Absolute_Primes.pdf

Proves (actually re-summarizes) that a n-digit permutable prime (excluding repunits) does not exist
for 3 < n < 6*10^175. In fact, the existence of large permutable primes is related to the distribution
of primes of which 10 is a primitive root.

( Note by Toshio Yamaguchi for my readers [ January 23, 2011 ]
   circular primes differ from permutable primes !
   The permutable primes are a subset of the circular primes.
   Circular primes: 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933
   Permutable primes: 2, 3, 5, 7, 13, 17, 37, 79, 113, 199, 337 )

David W. Wilson (email) reports the following after letting me know there are no circular primes of length 10, 11 and 12:
"On statistical grounds, I would be extremely surprised if any longer circular primes exist. A cursory analysis:"

Using the estimate of n/log(n) primes <= n, we can estimate that there are about 10^d/log(10^d)
d-digit primes (this estimate is on the high side, since it includes primes with fewer than d digits).
The probability that a d-digit number is prime is therefore about 1/log(10^d).

A d-digit circular prime other than a repunit must generate d distinct values by cycling.
By the above estimate, the probability that these d values will all be prime is about (1/log(10^d))^d,
and there should be about p = 10^d/(log(10^d)^d  d-digit circular primes.

For various d, the values of p are:

1   4.34294481903251827652
2   4.71529242529034823051
3   3.03381552720562897797
4   1.38962391600003339055
5   0.49439211408556231820
6   0.14381307270490575783
7   0.03538375334789643158
8   0.00754318214030964006
9   0.00141865311747126389
10  0.00023869488522460123

This argument is obviously very imprecise, but I think the general conclusion is valid,
namely that the expected number of d-digit non-repunit circular primes rapidly
approaches 0 as d increases.

On [ August 13, 2000 ] Walter Schneider (email), the author who determinded there
are no new circular primes of lengths 17, 18 and 19,
kindly wrote some explanatory notes thus clarifying the methods and tools he used.

"I think my program to search for circular primes is very obvious.
The main steps for circular primes of length n are:

1. Recursively generate all possible strings of length n.

Only the digits 1, 3, 7 and 9 have to be considered. I use strings
(not numbers) because this is much faster. Because we search only for the
least number in each cycle the recursion can be speeded up by some easy
tests. For example, digits 2 to n cannot be smaller than the first digit.
(Therefore most of the running time is spent when the first digit is one!)

2. For each string generated in 1. of length n determine the whole cycle
and test if the original string is the least one in the cycle. Note that
we still work with strings not numbers.

3. For each string in the cycle convert the string to a big integer and
test for a small factor. I use the LiDIA bigint library and make a
divisibility test by all primes less than 100.

4. Not too much numbers are left at this step. For each number in the cycle
make a fermat test.

As you can see there is nothing very special about my search program. One of the crucial facts
I think is the fast generation using strings not numbers and the quick test in step 3."




Sources Revealed

Enormous amounts of research have been performed on repunits.
Here's a reflection of them in a few websites :
bolRepunits Prime Factors
bolRepunit - E.W. Weisstein
bolREP-UNIT & REP-DIGIT

Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences

Base 10 entries about circular (cyclic) primes are the following :
%N Primes of form (10^n - 1)/9
        (next terms are for n = 317, 1031, 49081, 86453, 109297, 270343, 5794777, 8177207). under A004022.
%N (10^n - 1)/9 is prime (Indices of prime repunits). under A004023.
%N Circular primes (smallest element of numbers that remain prime under cyclic shifts of digits). under A016114.
%N Numbers such that every cyclic permutation is a prime. under A068652.
%N Circular primes that are not repunits. under A293663.
%N Numbers with nondecreasing digits such that every cyclic shift is a prime. under A263499.
%N Primes such that
        (a) reversal of digits gives a prime,
        (b) deleting any digit gives a prime, and
        (c) reversing digits and deleting any digit gives a prime. under A055604.
%N Palindromic primes that are "near-miss circular primes" (all digits alllowed). under A045978.
%N Near-miss circular primes (all digits alllowed). under A270083.

Base 10 entries about digit permutations are the following :
%N Every permutation of digits is a prime. under A003459.
%N Every permutation of digits is a prime with at least two different digits under A129338.
%N Every permutation of digits is a prime (smallest element). under A258706.

Base n entries about the subject are the following :
%N Largest nonrepunit base-n circular prime (conjectured). under A293142.
%N Largest nonrepunit base-n permutable prime (conjectured). under A317689.

Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.

Circular prime from “The Prime Glossary”.

Prime Curios! - site maintained by G. L. Honaker Jr. and Chris Caldwell
11 - The 1st prime repunit.
3304
193939
199933
285646799
(19 ones) 1111111111111111111 is a Repunit prime.
1111111111111111111
(23 ones) 11111111111111111111111 is a Repunit prime.
11111111111111111111111
317 - Number of digits in the 3rd prime repunit.
11111...11111 (317-digits)
The largest known Repunit prime has 1031 digits.
11111...11111 (1031-digits)


The Table


Circular Primes
Length 8177207
Total = ?		R8177207 Record Probable Prime Repunit, R(8177207)
Found by Serge Batalov and Ryan Propper [ May 8, 2021 ]
Length 5794777
Total = ?		R5794777 Probable Prime Repunit, R(5794777)
Found by Serge Batalov and Ryan Propper [ April 20, 2021 ]
Length 270343
Total = ?		R270343 Probable Prime Repunit, R(270343)
Found by Maksym Voznyy [ July 11, 2007 ]
Length 109297
Total = ?		R109297 Probable Prime Repunit, R(109297)
Found by Harvey Dubner [ March 28, 2007 ]
Length 86453
Total = ?		R86453 Probable Prime Repunit, R(86453)
Found by Lew Baxter [ October 26, 2000 ]
This repunit is now certified prime by Andreas Enge [ May 15, 2023 ]
Length 49081
Total = ?		R49081 Probable prime Repunit, R(49081)
Found by Harvey Dubner [ September, 1999 ]
This repunit is now certified prime by Paul Underwood [ March 21, 2022 ]
Length 1031
Total = ?		R1031 Repunit 1031
(10^1031-1)/9
Length 317
Total = ?		R317 Repunit 317
(10^317-1)/9
Length 25
Total = ?		 Unknown. ?
Length 24
Total = ?		 Unknown. ?
Length 23
Total = 1		R23 11111111111111111111111 is the only one !
Determined by W. Schneider & N. Pavlos
[ January 22, 2003 ]
Length 22
Total = 0		 Nihil !
Determined by Walter Schneider
[ December 13, 2002 ]
Length 21
Total = 0		 Nihil !
Determined by Walter Schneider
[ April 11, 2002 ]
Length 20
Total = 0		 Nihil !
Determined by Walter Schneider
[ April 11, 2002 ]
Length 19
Total = 1		R19 1111111111111111111 is the only one !
Determined by Walter Schneider
[ August 10, 2000 ]
Length 18
Total = 0		 Nihil !
Determined by Walter Schneider
[ July 28, 2000 ]
Length 17
Total = 0		 Nihil !
Determined by Walter Schneider
[ July 25, 2000 ]
Length 16
Total = 0		 Nihil ! Determined by Darren Smith
[ September 9, 1998 ]
Length 15
Total = 0		 Nihil ! Determined by Darren Smith
[ August 19, 1998 ]
Length 14
Total = 0		 Nihil ! Determined by Darren Smith
[ June 14, 1998 ]
Length 13
Total = 0		 Nihil ! Determined by Darren Smith
[ June 12, 1998 ]
Length 12
Total = 0		 Nihil !
Determined by David W. Wilson
[ February 18, 1998 ]
One Near Miss exists

     733793111393 prime
     337931113937 prime
     379311139373 prime
     793111393733 prime
     931113937337 prime
     311139373379 prime
     111393733793 prime
     113937337931 prime
     139373379311 prime
     393733793111 prime
     937337931113 prime
 but 373379311139 = 23 * 53 * 306299681
Length 11
Total = 0		 Nihil !
Determined by David W. Wilson
[ February 17, 1998 ]
Length 10
Total = 0		 Nihil !
Determined by David W. Wilson
[ February 17, 1998 ]
Length 9
Total = 0		 Nihil ! 
33191 eligible primes 

One Near Miss exists

     913311913 prime
     133119139 prime
     331191391 prime
     311913913 prime
     119139133 prime
     191391331 prime
     913913311 prime
     139133119 prime
 but 391331191 = 29 * 131 * 239 * 431
Length 8
Total = 0		 Nihil ! 
9177 eligible primes 

One Near Miss exists

     71777393 prime
     17773937 prime
     77739371 prime
     77393717 prime
     73937177 prime
     39371777 prime
     93717773 prime
 but 37177739 = 29 * 683 * 1877
Length 7
Total = 0		 Nihil ! 
2709 eligible primes

Two Near Misses exist

a)   9197777 prime
     1977779 prime
     9777791 prime
     7777919 prime
     7779197 prime
     7791977 prime
 but 7919777 = 83 * 95419

b)   9991313 prime
     9913139 prime
     9131399 prime
     1313999 prime
     3139991 prime
     1399913 prime
 but 3999131 = 17 * 235243
Length 6
Total = 2		 193939
199933 
757 eligible primes
Length 5
Total = 2		 11939
19937 
249 eligible primes
Length 4
Total = 2		 1193
3779 
63 eligible primes
Length 3
Total = 4		 113
197
199
337 
30 eligible primes
Length 2
Total = 5		 11 = R2
13
17
37
79 
10 eligible primes
Length 1
Total = 5		 (1)
2 *
3
5 *
7 
3 eligible primes
4 if 1 is taken as a prime

( apart from 2 and 5 )




Circular & Permutable Primes in bases b

Table with circular primes in other bases
Presented by Xinyao Chen

(repunit primes excluded, since all repunit primes are circular primes, and a heuristic argument is that
there are infinitely many repunit primes in every non-perfect power bases {the repunits in perfect power
bases can be factored algebraically, thus there is at most one repunit prime in these bases}).

 Circular Primes in other bases b (b ≤ 12) and Repunits excluded
(using A−Z to represent digit values 10 to 35)
Base 2none exist
Base 32, 12, 21
Base 413, 31, 113, 131, 311, 11333, 13331, 31133, 33113, 33311
Base 512, 21, 23, 32, 34, 43, 1132, 1321, 1424, 2113, 2414, 3211, 4142, 4241, 13234, 14444, 23413, 32341, 34132, 41323, 41444, 44144, 44414, 44441
Base 615, 51, 155, 515, 551
Base 72, 3, 5, 14, 16, 23, 25, 32, 41, 52, 56, 61, 65, 142, 155, 166, 214, 245, 421, 452, 515, 524, 551, 616, 661, 1165, 1253, 1325, 1444, 1543, 1651, 1655, 2326, 2513, 2531, 2564, 2623, 2656, 3125, 3154, 3251, 3262, 4144, 4256, 4315, 4414, 4441, 4555, 5116, 5132, 5165, 5312, 5431, 5455, 5516, 5545, 5554, 5626, 5642, 6232, 6265, 6425, 6511, 6551, 6562, 11515, 15115, 15151, 15652, 21565, 36644, 43664, 44366, 51151, 51511, 52156, 56521, 64436, 65215, 66443, 125452, 141544, 154414, 212545, 254521, 414154, 415441, 441415, 452125, 521254, 544141, 545212, 1255165, 1651255, 2546455, 2551651, 4552546, 4645525, 5125516, 5165125, 5254645, 5464552, 5516512, 5525464, 6455254, 6512551, 13416163, 14445665, 16163134, 16313416, 31341616, 34161631, 41616313, 44456651, 44566514, 45665144, 51444566, 56651444, 61631341, 63134161, 65144456, 66514445, 1216414336, 1433612164, 1641433612, 2164143361, 3361216414, 3612164143, 4143361216, 4336121641, 6121641433, 6414336121, 124464346432, 212446434643, 244643464321, 321244643464, 346432124464, 432124464346, 434643212446, 446434643212, 464321244643, 464346432124, 643212446434, 643464321244
Base 82, 3, 5, 7, 15, 35, 37, 51, 53, 57, 73, 75, 1137, 1317, 1357, 1371, 1713, 1775, 3171, 3337, 3373, 3571, 3711, 3733, 5177, 5713, 7113, 7131, 7135, 7333, 7517, 7751, 137717, 155753, 171377, 315575, 377171, 531557, 557531, 575315, 713771, 717137, 753155, 771713, 17575733, 31757573, 33175757, 57331757, 57573317, 73317575, 75733175, 75757331
Base 92, 3, 5, 7, 12, 14, 18, 21, 25, 41, 47, 52, 74, 78, 81, 87, 122, 175, 212, 221, 254, 278, 425, 517, 542, 751, 782, 788, 827, 878, 887, 1288, 1477, 1772, 1857, 2177, 2285, 2852, 2881, 4555, 4771, 5228, 5455, 5545, 5554, 5718, 7147, 7185, 7217, 7714, 7721, 7778, 7787, 7877, 8128, 8522, 8571, 8777, 8812, 12815, 15128, 15272, 21527, 27215, 28151, 51281, 52721, 55777, 57775, 72152, 75577, 77557, 77755, 81512, 121278, 124147, 127812, 128844, 144574, 144745, 145582, 147124, 181872, 187218, 212781, 214558, 218187, 224557, 227824, 241471, 242278, 245572, 247754, 274457, 278121, 278242, 288441, 412884, 414457, 414712, 422782, 424775, 441288, 445727, 445741, 447451, 451447, 455722, 455821, 457274, 457414, 457577, 471241, 474514, 477542, 514474, 542477, 557224, 558214, 572245, 572744, 574144, 575774, 577457, 582145, 712414, 721818, 722455, 727445, 741445, 744572, 745144, 745757, 754247, 757745, 774575, 775424, 781212, 782422, 812127, 818721, 821455, 824227, 844128, 872181, 884412, 5758778, 5877857, 7587785, 7785758, 7857587, 8575877, 8778575
Base 102, 3, 5, 7, 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
Base 11
2, 3, 5, 7, 12, 16, 18, 21, 27, 29, 34, 3A, 43, 49, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, A3, 117, 139, 171, 193, 1AA, 236, 269, 319, 335, 353, 362, 364, 36A, 391, 3A6, 436, 533, 566, 588, 623, 63A, 643, 656, 665, 692, 6A3, 711, 7AA, 858, 885, 913, 926, 931, A1A, A36, A63, A7A, AA1, AA7, 1244, 124A, 1459, 1569, 1574, 1875, 2441, 24A1, 2597, 2766, 2777, 2795, 2948, 34A4, 3538, 3574, 37A7, 3835, 3A6A, 4124, 4157, 434A, 4357, 4412, 44A9, 4591, 4779, 478A, 4829, 4A12, 4A43, 4A94, 5187, 5279, 5383, 5691, 5741, 5743, 57A9, 5914, 5972, 6276, 6627, 6915, 698A, 6A3A, 7259, 7277, 737A, 7415, 7435, 7518, 7662, 7727, 7772, 7794, 78A4, 7947, 7952, 7A73, 7A95, 8294, 8353, 8751, 8A47, 8A69, 9145, 9156, 944A, 9477, 9482, 9527, 957A, 9725, 98A6, 9AAA, A124, A3A6, A434, A478, A698, A6A3, A737, A944, A957, A9AA, AA9A, AAA9, 1989A, 1A989, 36774, 43677, 67743, 74367, 77436, 7AAAA, 891A9, 89A19, 91A98, 9891A, 989A1, 9A198, A1989, A7AAA, A9891, AA7AA, AAA7A, AAAA7, 111743, 112157, 112465, 115174, 115334, 116364, 117431, 121571, 1217A8, 124651, 125366, 125515, 12AA64, 132263, 134777, 136977, 137165, 139347, 139644, 141429, 141515, 142914, 14426A, 144695, 145782, 145A92, 146228, 147674, 14A882, 151255, 151415, 151514, 151741, 153341, 153372, 155336, 157112, 157169, 157989, 157A4A, 15A755, 162956, 163641, 164226, 165137, 16543A, 167663, 167829, 168862, 169157, 174115, 174311, 174933, 177A75, 17A812, 182198, 18264A, 184365, 188965, 18A783, 18AA86, 198182, 198892, 199479, 199486, 19A975, 1A2736, 1A29A7, 1A6664, 1A7885, 1A9692, 214578, 2145A9, 214A88, 215337, 215711, 216886, 217A81, 219818, 219889, 21A969, 226164, 226313, 228146, 228AA9, 232A57, 235247, 238794, 23967A, 242A94, 244289, 246511, 247235, 249877, 25347A, 253661, 255151, 2585A3, 261642, 263132, 264975, 264A18, 267A75, 26A144, 27332A, 27361A, 279779, 27A336, 27A868, 281462, 289244, 289729, 28AA92, 291414, 291678, 292897, 2945A3, 295445, 295537, 295616, 29A71A, 2A2733, 2A2A34, 2A342A, 2A5723, 2A9424, 2A99A3, 2AA641, 311174, 313226, 316766, 317493, 318A78, 322631, 32585A, 32945A, 32A273, 32A572, 32A99A, 331749, 332A27, 334115, 336155, 33627A, 337215, 341153, 342A2A, 347139, 347771, 347A25, 352472, 354498, 35A959, 361553, 361A27, 3627A3, 364116, 364A97, 365184, 366125, 367845, 369771, 371651, 372153, 372955, 384477, 387942, 38A956, 393471, 396441, 3967A2, 397855, 39A955, 3A1654, 3A3A87, 3A873A, 411517, 411533, 411636, 412AA6, 413964, 414291, 414767, 415151, 41A666, 422616, 423879, 4242A9, 426A14, 428924, 429141, 42A2A3, 42A942, 431117, 436518, 43A165, 441396, 4426A1, 442892, 445295, 446951, 447738, 449835, 452954, 453678, 455775, 457821, 4584A8, 45A329, 45A921, 462281, 465112, 467895, 469514, 46A788, 471393, 472352, 476741, 477384, 477713, 479199, 47954A, 479849, 47A253, 484979, 486199, 48865A, 48A4A5, 493317, 494798, 497526, 497948, 498354, 498772, 4A157A, 4A1826, 4A4795, 4A548A, 4A8458, 4A8821, 4A9736, 511246, 512551, 513716, 514151, 514469, 515125, 515141, 515A75, 517411, 5177A7, 518436, 518896, 519A97, 51A788, 524723, 526497, 5267A7, 529544, 533411, 533615, 533721, 5347A2, 536612, 536784, 537295, 539785, 539A95, 543A16, 544529, 544983, 545577, 546789, 548A4A, 54A479, 551512, 5515A7, 553361, 553729, 553978, 5539A9, 556986, 5576AA, 557754, 561629, 5638A9, 569865, 571121, 571691, 57232A, 575A88, 576AA5, 577545, 577699, 578214, 579891, 57A4A1, 584A84, 585A32, 5935A9, 5A3258, 5A3294, 5A4886, 5A7551, 5A8857, 5A9214, 5A9593, 5A9AA9, 612536, 615533, 616295, 616422, 618AA8, 619948, 61A273, 621688, 622814, 627A33, 629561, 631322, 631676, 636411, 638A95, 641163, 6412AA, 641A66, 642261, 644139, 649752, 64A182, 64A973, 651124, 651371, 651843, 651889, 6543A1, 655698, 65A488, 661253, 663167, 6641A6, 66641A, 674147, 676631, 678291, 678453, 678954, 67A239, 67A752, 6827A8, 688621, 691571, 6921A9, 695144, 697713, 698655, 699577, 6A1442, 6A7884, 6A86A9, 6A96A8, 6AA557, 711215, 713477, 713697, 713934, 716513, 716915, 71A29A, 721533, 7232A5, 723524, 724987, 729289, 729553, 7332A2, 7361A2, 7364A9, 738447, 73A3A8, 741151, 741476, 743111, 749331, 75177A, 7519A9, 752649, 75267A, 754557, 75515A, 75A885, 766316, 767414, 769957, 76AA55, 771347, 771369, 772498, 773844, 775455, 776995, 777134, 779279, 77A751, 782145, 782916, 78318A, 784536, 785539, 78846A, 78851A, 789546, 791994, 792797, 794238, 794849, 7954A4, 797792, 798494, 798915, 7A2396, 7A2534, 7A3362, 7A4A15, 7A7517, 7A7526, 7A8121, 7A8682, 81217A, 814622, 818219, 821457, 8214A8, 821981, 8264A1, 827A86, 829167, 8318A7, 835449, 843651, 844773, 845367, 84584A, 846A78, 849479, 849794, 84A845, 851A78, 855397, 8575A8, 85A325, 8618AA, 861994, 862168, 865569, 865A48, 86827A, 86A96A, 873A3A, 877249, 879423, 88214A, 8846A7, 8851A7, 88575A, 886216, 8865A4, 889219, 889651, 891579, 892198, 892442, 895467, 896518, 897292, 8A4A54, 8A7831, 8A9563, 8AA861, 8AA922, 914142, 915716, 915798, 916782, 919947, 92145A, 921988, 921A96, 9228AA, 924428, 927977, 928972, 933174, 934713, 935A95, 942387, 94242A, 945A32, 947919, 947984, 948497, 948619, 951446, 954452, 954678, 954A47, 955372, 95539A, 956162, 95638A, 957769, 95935A, 95A9AA, 964413, 965188, 967A23, 96921A, 96A86A, 972928, 97364A, 97519A, 975264, 977136, 977927, 978553, 979484, 981821, 983544, 984947, 986556, 987724, 988921, 989157, 994791, 994861, 995776, 99A32A, 9A32A9, 9A71A2, 9A9553, 9A9751, 9AA95A, A14426, A157A4, A16543, A18264, A23967, A25347, A27332, A27361, A29A71, A2A342, A32585, A32945, A32A99, A33627, A342A2, A3A873, A47954, A48865, A4A157, A4A548, A548A4, A5576A, A57232, A6412A, A66641, A71A29, A75177, A75267, A75515, A78318, A78846, A78851, A81217, A84584, A8618A, A86827, A86A96, A873A3, A88214, A88575, A92145, A9228A, A94242, A95539, A95638, A95935, A95A9A, A96921, A96A86, A97364, A97519, A99A32, A9AA95, AA5576, AA6412, AA8618, AA9228, AA95A9, 1156275A, 11754793, 13374526, 13441815, 14357845, 15134418, 156275A1, 16287845, 1744368A, 17547931, 18151344, 18823496, 188384A7, 19743492, 19784657, 1A523AA2, 21974349, 21A523AA, 22496A8A, 23496188, 23AA21A5, 2496A8A2, 25A6A527, 26133745, 2678729A, 2725A6A5, 275A1156, 28784516, 29478643, 29A26787, 31175479, 32947864, 3356444A, 33745261, 34418151, 34921974, 34961882, 356444A3, 35784514, 36657969, 36895738, 368A1744, 37452613, 37693939, 37A49644, 38368957, 38396479, 384A7188, 39376939, 39393769, 39647938, 3AA21A52, 41815134, 43294786, 43492197, 43578451, 4368A174, 437A4964, 44181513, 44368A17, 4437A496, 444A3356, 44A33564, 45143578, 45162878, 45261337, 45486987, 46571978, 47864329, 47931175, 47938396, 48698745, 49219743, 49618823, 4964437A, 496A8A22, 4A335644, 4A718838, 51344181, 51435784, 51628784, 523AA21A, 52613374, 52725A6A, 54793117, 54869874, 56275A11, 56444A33, 57197846, 57383689, 57845143, 57969366, 5A115627, 5A6A5272, 61337452, 61882349, 6275A115, 62878451, 64329478, 64437A49, 6444A335, 64793839, 65719784, 65796936, 66579693, 678729A2, 68957383, 68A17443, 69366579, 69393937, 69874548, 6A52725A, 6A8A2249, 7188384A, 71978465, 725A6A52, 729A2678, ..., AA657365177398
Base 122, 3, 5, 7, B, 15, 51, 57, 5B, 75, B5, 117, 11B, 171, 175, 1B1, 1B7, 517, 711, 71B, 751, B11, B71, 157B, 555B, 55B5, 57B1, 5B55, 7B15, B157, B555, 115B77, 15B771, 5B7711, 7115B7, 77115B, B77115
The datas for bases 2 to 6 are known to be complete, but the datas for bases 7 to 12 are only conjectured to be complete, there is still no proof that these datas are really complete.


Table with permutable primes in other bases
Presented by Xinyao Chen

(repunit primes excluded, since all repunit primes are permutable primes, and a heuristic argument is that
there are infinitely many repunit primes in every non-perfect power bases {the repunits in perfect power
bases can be factored algebraically, thus there is at most one repunit prime in these bases}).

 Permutable Primes in other bases b (b ≤ 36) and Repunits excluded
(using A−Z to represent digit values 10 to 35)
Base 2none exist
Base 32, 12, 21
Base 42, 3, 13, 31, 113, 131, 311
Base 52, 3, 12, 21, 23, 32, 34, 43, 14444, 41444, 44144, 44414, 44441
Base 62, 3, 5, 15, 51, 155, 515, 551
Base 72, 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, ...
Base 82, 3, 5, 7, 15, 35, 37, 51, 53, 57, 73, 75, 3337, 3373, 3733, 7333, ...
Base 92, 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, ...
Base 102, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, ...
Base 112, 3, 5, 7, 12, 16, 18, 21, 27, 29, 34, 3A, 43, 49, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, A3, 117, 139, 171, 193, 1AA, 319, 335, 353, 36A, 391, 3A6, 533, 566, 588, 63A, 656, 665, 6A3, 711, 7AA, 858, 885, 913, 931, A1A, A36, A63, A7A, AA1, AA7, 2777, 7277, 7727, 7772, 9AAA, A9AA, AA9A, AAA9, 7AAAA, A7AAA, AA7AA, AAA7A, AAAA7, ...
Base 122, 3, 5, 7, B, 15, 51, 57, 5B, 75, B5, 117, 11B, 171, 1B1, 711, B11, 555B, 55B5, 5B55, B555, ...
Base 132, 3, 5, 7, B, 14, 16, 1A, 23, 25, 32, 38, 41, 52, 56, 58, 61, 65, 6B, 7A, 7C, 83, 85, 9A, A1, A7, A9, B6, C7, 11B, 133, 155, 1B1, 229, 247, 274, 292, 313, 331, 33B, 388, 3B3, 427, 472, 515, 551, 724, 742, 779, 78A, 797, 7A8, 838, 87A, 883, 8A7, 922, 977, A78, A87, B11, B33, 4445, 4454, 4544, 5444, 6667, 6676, 6766, 7666, ...
Base 142, 3, 5, 7, B, D, 13, 15, 19, 31, 35, 3B, 51, 53, 59, 91, 95, 9B, 9D, B3, B9, BD, D9, DB, 33D, 3D3, D33, 1119, 1191, 1911, 9111, ...
Base 152, 3, 5, 7, B, D, 12, 14, 1E, 21, 27, 2B, 2D, 41, 47, 4D, 72, 74, 78, 87, 8B, B2, B8, D2, D4, E1, 1DD, 227, 272, 44B, 4B4, 722, 77D, 7D7, B44, BEE, D1D, D77, DD1, EBE, EEB, 1444, 4144, 4414, 4441, 7777D, 777D7, 77D77, 7D777, D7777, ...
Base 162, 3, 5, 7, B, D, 17, 1F, 35, 3B, 3D, 53, 59, 71, 95, B3, BF, D3, F1, FB, 115, 11B, 151, 1B1, 377, 511, 55D, 5D5, 737, 773, 7BB, B11, B7B, BB7, BDD, D55, DBD, DDB, ...
Base 172, 3, 5, 7, B, D, 16, 1E, 23, 2D, 32, 38, 3A, 45, 4B, 54, 5G, 61, 6B, 7C, 83, 8D, 8F, 9A, A3, A9, AB, B4, B6, BA, C7, D2, D8, E1, F8, G5, 7AA, 7GG, A7A, AA7, G7G, GG7, 6DDD, D6DD, DD6D, DDD6, ...
Base 182, 3, 5, 7, B, D, H, 1B, 57, 5D, 5H, 75, 7D, B1, D5, D7, H5, 11B, 11H, 1B1, 1DD, 1H1, 55B, 5B5, 7BB, 7HH, B11, B55, B7B, BB7, BBH, BHB, D1D, DD1, H11, H7H, HBB, HH7, BDDD, DBDD, DDBD, DDDB, ...
Base 192, 3, 5, 7, B, D, H, 1A, 1C, 23, 25, 29, 32, 34, 3A, 3E, 3G, 43, 47, 4D, 52, 58, 5C, 5E, 5I, 74, 7G, 7I, 85, 8F, 92, 9A, A1, A3, A9, BE, BI, C1, C5, D4, DG, E3, E5, EB, EH, F8, G3, G7, GD, HE, I5, I7, IB, 113, 122, 131, 1CC, 1FF, 212, 221, 22D, 29E, 2D2, 2E9, 311, 377, 3AA, 44B, 4B4, 737, 773, 77H, 7H7, 92E, 9E2, A3A, AA3, B44, BFF, C1C, CC1, D22, E29, E92, F1F, FBF, FF1, FFB, H77, 2DDD, D2DD, DD2D, DDD2, ...
Base 202, 3, 5, 7, B, D, H, J, 13, 19, 31, 3B, 3D, 3J, 7B, 7H, 91, 9B, 9D, 9H, 9J, B3, B7, B9, BD, D3, D9, DB, DH, H7, H9, HD, HJ, J3, J9, JH, 33H, 3H3, 77D, 7D7, D77, H33, 1113, 1119, 1131, 1191, 1311, 1911, 3111, 9111, ...
Base 212, 3, 5, 7, B, D, H, J, 12, 1A, 1G, 1K, 21, 25, 2B, 2H, 2J, 45, 4D, 52, 54, 58, 85, 8B, 8D, A1, AD, AH, AJ, B2, B8, BK, D4, D8, DA, DK, G1, GH, H2, HA, HG, J2, JA, JK, K1, KB, KD, KJ, 15B, 188, 1AA, 1B5, 1JJ, 1KK, 28D, 2D8, 51B, 5B1, 5BB, 818, 82D, 881, 8D2, A1A, AA1, B15, B51, B5B, BB5, D28, D82, J1J, JJ1, K1K, KK1, 111G, 11G1, 1G11, 444B, 44B4, 4B44, B444, G111, ...
Base 222, 3, 5, 7, B, D, H, J, 19, 1F, 1J, 1L, 35, 37, 53, 5H, 5L, 73, 7D, 91, D7, F1, FH, FJ, H5, HF, J1, JF, L1, L5, 155, 1FF, 33D, 3D3, 515, 551, 55J, 5J5, 77H, 7H7, 9JJ, D33, DFF, F1F, FDF, FF1, FFD, FFH, FHF, H77, HFF, J55, J9J, JJ9, 333H, 33H3, 3H33, H333, ...
Base 232, 3, 5, 7, B, D, H, J, 16, 1K, 27, 2F, 3A, 3K, 49, 4B, 4F, 4L, 5C, 5G, 61, 6J, 72, 7C, 7I, 7K, 8D, 8F, 94, A3, AB, B4, BA, BG, C5, C7, D8, EF, F2, F4, F8, FE, FM, G5, GB, GL, HI, I7, IH, J6, JK, K1, K3, K7, KJ, L4, LG, MF, 133, 166, 1AA, 313, 331, 33D, 3D3, 49I, 4I9, 616, 661, 66H, 6EF, 6FE, 6H6, 94I, 9I4, A1A, AA1, BBH, BHB, D33, DII, DJJ, E6F, EF6, F6E, FE6, H66, HBB, I49, I94, IDI, IID, JDJ, JJD, 2999, 9299, 9929, 9992, BCCC, BIII, CBCC, CCBC, CCCB, IBII, IIBI, IIIB, ...
Base 242, 3, 5, 7, B, D, H, J, N, 1D, 1H, 1J, 57, 5B, 5J, 75, 7B, B5, B7, BH, BJ, D1, H1, HB, HN, J1, J5, JB, JN, NH, NJ, 155, 515, 551, 77H, 7H7, H77, ...
Base 252, 3, 5, 7, B, D, H, J, N, 14, 16, 1G, 29, 2B, 2N, 34, 3E, 41, 43, 47, 49, 61, 67, 6D, 6H, 74, 76, 7I, 7M, 7O, 8B, 92, 94, 9E, 9G, B2, B8, BI, CD, D6, DC, DM, DO, E3, E9, EH, G1, G9, GJ, GL, H6, HE, HI, HO, I7, IB, IH, JG, JO, LG, LM, M7, MD, ML, N2, O7, OD, OH, OJ, 113, 11N, 122, 131, 18E, 199, 1BB, 1E8, 1N1, 1NN, 212, 221, 22L, 289, 298, 2BO, 2GJ, 2JG, 2L2, 2OB, 311, 3EE, 77B, 7B7, 7OO, 81E, 829, 892, 8E1, 919, 928, 982, 991, 9HH, B1B, B2O, B77, BB1, BO2, CCH, CHC, E18, E3E, E81, EE3, G2J, GJ2, H9H, HCC, HH9, IIJ, IJI, J2G, JG2, JII, L22, MMN, MNM, N11, N1N, NMM, NN1, O2B, O7O, OB2, OO7, JMMM, MJMM, MMJM, MMMJ, ...
Base 262, 3, 5, 7, B, D, H, J, N, 13, 15, 1H, 1L, 31, 3N, 3P, 51, 59, 5J, 79, 7B, 7F, 7H, 95, 97, 9N, B7, BL, BP, F7, FJ, H1, H7, HL, J5, JF, L1, LB, LH, LN, N3, N9, NL, P3, PB, 117, 11P, 171, 1P1, 335, 337, 353, 373, 533, 5BB, 5LL, 711, 733, 77J, 7J7, B5B, BB5, J77, L5L, LL5, LLP, LPL, P11, PLL, ...
Base 272, 3, 5, 7, B, D, H, J, N, 14, 1A, 1E, 1G, 1K, 25, 27, 2D, 2H, 2P, 41, 45, 52, 54, 5E, 5M, 72, 78, 7A, 7M, 87, 8D, 8H, 8P, A1, A7, AB, AN, BA, BE, BG, D2, D8, DM, E1, E5, EB, G1, GB, GP, H2, H8, HK, K1, KH, KN, M5, M7, MD, MN, NA, NK, NM, P2, P8, PG, PQ, QP, 1AE, 1EA, 1KK, 22J, 2J2, 77H, 7H7, A1E, AE1, E1A, EA1, H77, HMM, J22, K1K, KK1, MHM, MMH, 2225, 2252, 2522, 5222, ...
Base 282, 3, 5, 7, B, D, H, J, N, 1F, 1P, 3D, 3H, 3N, 59, 5B, 5R, 95, 9B, 9J, 9P, B5, B9, D3, DF, F1, FD, FJ, FN, H3, HN, HR, J9, JF, JP, N3, NF, NH, P1, P9, PJ, R5, RH, 1BB, 33N, 3DD, 3N3, 55D, 5D5, B1B, BB1, BDD, D3D, D55, DBD, DD3, DDB, FFH, FHF, HFF, JJN, JNJ, JRR, N33, NJJ, NPP, PNP, PPN, RJR, RRJ, ...
Base 292, 3, 5, 7, B, D, H, J, N, 12, 18, 1C, 1I, 21, 23, 29, 2D, 2P, 32, 3A, 3E, 3G, 3M, 3Q, 4F, 4L, 56, 5C, 5M, 65, 6H, 6J, 6N, 78, 7K, 7Q, 81, 87, 89, 8P, 92, 98, 9M, A3, AH, AL, AN, BC, BS, C1, C5, CB, CJ, D2, DK, DO, E3, EF, EP, ER, F4, FE, FM, FQ, FS, G3, GN, H6, HA, HS, I1, IJ, IP, J6, JC, JI, JK, JQ, K7, KD, KJ, L4, LA, LM, M3, M5, M9, MF, ML, N6, NA, NG, NO, OD, ON, P2, P8, PE, PI, Q3, Q7, QF, QJ, RE, RS, SB, SF, SH, SR, 16O, 177, 1AM, 1MA, 1O6, 3DR, 3RD, 4GJ, 4JG, 61O, 6KR, 6O1, 6RK, 717, 771, 7DD, 7II, 7OO, 88H, 8H8, A1M, AM1, CCH, CHC, D3R, D7D, DD7, DDJ, DFF, DJD, DR3, EEH, EHE, FDF, FFD, G4J, GJ4, H88, HCC, HEE, HKK, I7I, II7, J4G, JDD, JG4, JOO, K6R, KHK, KKH, KR6, M1A, MA1, O16, O61, O7O, OJO, OO7, OOJ, R3D, R6K, RD3, RK6, 444F, 44F4, 4F44, F444, ...
Base 302, 3, 5, 7, B, D, H, J, N, T, 17, 1B, 1N, 71, 7D, 7J, 7T, B1, BH, BN, BT, D7, DT, HB, J7, JN, N1, NB, NJ, T7, TB, TD, 11B, 1B1, 1NN, 7DD, 7HH, B11, BBD, BDB, BDD, BTT, D7D, DBB, DBD, DD7, DDB, DJJ, H7H, HH7, JDJ, JJD, N1N, NN1, TBT, TTB, ...
Base 312, 3, 5, 7, B, D, H, J, N, T, 1A, 1C, 1M, 25, 29, 2L, 2R, 34, 38, 3A, 3G, 43, 52, 5I, 5Q, 67, 6B, 6D, 76, 7A, 7C, 7G, 7O, 83, 8L, 8T, 92, 9E, 9S, A1, A3, A7, AL, B6, BC, BI, C1, C7, CB, CP, CT, D6, DG, DI, DS, E9, EF, EN, FE, FQ, G3, G7, GD, GR, HU, I5, IB, ID, IJ, JI, JS, KN, KR, L2, L8, LA, LQ, M1, MR, NE, NK, NQ, NU, O7, PC, Q5, QF, QL, QN, R2, RG, RK, RM, S9, SD, SJ, T8, TC, UH, UN, 11H, 11T, 188, 199, 1H1, 1NN, 1T1, 229, 22F, 292, 2F2, 55R, 5CC, 5R5, 7DT, 7QQ, 7TD, 818, 881, 919, 922, 991, 99N, 9GG, 9N9, BKM, BMK, C5C, CC5, D7T, DT7, EEP, EPE, F22, FPR, FRP, G9G, GG9, H11, HHJ, HJH, JHH, KBM, KMB, MBK, MKB, N1N, N99, NN1, NOO, ONO, OON, OPU, OUP, PEE, PFR, POU, PRF, PUO, Q7Q, QQ7, R55, RFP, RPF, T11, T7D, TD7, UOP, UPO, 555Q, 55Q5, 5Q55, 777M, 77M7, 7M77, M777, Q555, 14444, 41444, 44144, 44414, 44441, GGGGP, GGGPG, GGPGG, GPGGG, PGGGG, ...
Base 322, 3, 5, 7, B, D, H, J, N, T, V, 1B, 1L, 1T, 35, 37, 3D, 3H, 53, 57, 5D, 5J, 5L, 5V, 73, 75, 7F, 9J, 9P, 9T, B1, BF, BL, D3, D5, DH, DR, F7, FB, FN, H3, HD, HR, J5, J9, L1, L5, LB, NF, NP, P9, PN, PT, RD, RH, T1, T9, TP, V5, 33D, 3D3, 55H, 5H5, 7PP, D33, H55, P7P, PP7, BDDD, DBDD, DDBD, DDDB, FVVV, RVVV, VFVV, VRVV, VVFV, VVRV, VVVF, VVVR, ...
Base 332, 3, 5, 7, B, D, H, J, N, T, V, 1A, 1E, 1K, 1Q, 25, 27, 2D, 2H, 2N, 4J, 4P, 52, 58, 5E, 5Q, 5S, 5W, 72, 78, 7A, 7W, 85, 87, 8H, A1, A7, AH, AN, AT, D2, DK, DS, DW, E1, E5, EP, ET, GJ, H2, H8, HA, J4, JG, JQ, K1, KD, N2, NA, NS, P4, PE, Q1, Q5, QJ, QT, S5, SD, SN, TA, TE, TQ, W5, W7, WD, 227, 22T, 272, 2T2, 55D, 5D5, 722, 88V, 8V8, AAJ, AJA, ANW, AWN, D55, EET, ETE, JAA, NAW, NWA, T22, TEE, V88, WAN, WNA, 111S, 11S1, 1S11, 5SSS, 777G, 77G7, 7G77, G777, S111, S5SS, SS5S, SSS5, ...
Base 342, 3, 5, 7, B, D, H, J, N, T, V, 13, 17, 19, 1D, 1J, 1R, 1X, 31, 35, 37, 3P, 53, 59, 5B, 5L, 5N, 5T, 71, 73, 7D, 7J, 7P, 7V, 7X, 91, 95, 9B, 9P, 9V, B5, B9, BF, BR, D1, D7, DF, DJ, DL, DP, FB, FD, FV, J1, J7, JD, JR, L5, LD, N5, NR, NT, P3, P7, P9, PD, R1, RB, RJ, RN, RT, T5, TN, TR, TX, V7, V9, VF, VX, X1, X7, XT, XV, 33N, 3BB, 3N3, 5JN, 5NJ, 77X, 7X7, 99J, 9BB, 9J9, 9PP, B3B, B9B, BB3, BB9, BLX, BXL, DDN, DDR, DLV, DND, DRD, DVL, FRT, FTR, J5N, J99, JN5, LBX, LDV, LVD, LXB, N33, N5J, NDD, NJ5, P9P, PP9, RDD, RFT, RTF, TFR, TRF, VDL, VLD, X77, XBL, XLB, 1JJJ, 5777, 7577, 7757, 7775, 7DDD, D7DD, DD7D, DDD7, J1JJ, JJ1J, JJJ1, JPPP, PJPP, PPJP, PPPJ, ...
Base 352, 3, 5, 7, B, D, H, J, N, T, V, 12, 16, 18, 1C, 1I, 1Q, 21, 23, 29, 2D, 2R, 2V, 32, 38, 3M, 3W, 3Y, 4B, 4H, 4N, 61, 6D, 6H, 6N, 6T, 6V, 81, 83, 8D, 8R, 8V, 8X, 92, 9G, 9W, B4, BC, BG, BY, C1, CB, CD, CJ, D2, D6, D8, DC, DO, G9, GB, GX, H4, H6, HI, HM, HO, I1, IH, IN, IT, IV, JC, JQ, M3, MH, MR, N4, N6, NI, NO, NY, OD, OH, ON, Q1, QJ, QR, R2, R8, RM, RQ, T6, TI, V2, V6, V8, VI, VW, W3, W9, WV, WX, X8, XG, XW, Y3, YB, YN, 11V, 19R, 1CC, 1DD, 1R9, 1V1, 22T, 2T2, 33D, 33N, 36Y, 3D3, 3N3, 3Y6, 63Y, 6JO, 6OJ, 6Y3, 88B, 88N, 8B8, 8N8, 91R, 99D, 9D9, 9R1, B88, BBH, BHB, BQQ, BRR, C1C, CC1, D1D, D33, D99, DD1, GJM, GMJ, HBB, J6O, JGM, JMG, JO6, JRR, JVV, JYY, MGJ, MJG, MMV, MVM, N33, N88, NWW, NXX, O6J, OJ6, OOV, OVO, QBQ, QQB, R19, R91, RBR, RJR, RRB, RRJ, T22, V11, VJV, VMM, VOO, VVJ, VXX, WNW, WWN, XNX, XVX, XXN, XXV, Y36, Y63, YJY, YYJ, 222J, 22J2, 2J22, J222, MMMT, MMTM, MTMM, TMMM, ...
Base 362, 3, 5, 7, B, D, H, J, N, T, V, 15, 1B, 1H, 1N, 1V, 51, 5B, 5H, 7H, 7J, 7P, 7T, 7V, B1, B5, BD, BN, BP, DB, DV, H1, H5, H7, HJ, HT, HZ, J7, JH, JP, JZ, N1, NB, NZ, P7, PB, PJ, PT, T7, TH, TP, V1, V7, VD, VZ, ZH, ZJ, ZN, ZV, 155, 1JN, 1NJ, 515, 551, 5DD, 77D, 7D7, BBV, BDD, BHJ, BJH, BVB, D5D, D77, DBD, DD5, DDB, DDZ, DJJ, DZD, DZZ, HBJ, HJB, J1N, JBH, JDJ, JHB, JJD, JJV, JN1, JVJ, N1J, NJ1, VBB, VJJ, ZDD, ZDZ, ZZD, BBBT, BBTB, BTBB, TBBB, ...
In base 10 and base 12, as well as all bases ≤ 10 and all even bases ≤ 32, every permutable prime is a repunit or a near-repdigit, i.e. 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).
Theorem: 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 n x's) must be divisible by p. Since p does not divide x, so p must divide the repunit with n 1's. 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).



Contributions and Messages

David W. Wilson (email) determined there are no circular primes for lengths 10, 11 and 12.

Darren Smith (email) from Milwaukee, Wisconsin determined there are no circular primes for lengths 13, 14, 15 and 16.

Walter Schneider (email) determined there are no circular primes for length 17.
The total running time on his Pentium-III 550 Mhz was 14 hours.
and three days later determined also none for length 18.
Total running time was 2 days.
Finally, [ August 10, 2000 ]  Walter determined that Repunit R19 is the only circular prime of length 19.

[ April 11, 2002 ]
Walter Schneider (email) improved his search program for circular primes
considerably and used it to search for circular primes of
digit length 20 and 21. No solutions were found.
Running time on a Pentium-3 550Mhz was 19 hours for 20 digits
and a couple of days for 21 digits.

[ July 11, 2008 ]
Justin Chan (email) makes an important contribution ! - go to topic

[ April 20, 2022 ]
Message forwarded to me by Xinyao Chen (email).
The repunit R(49081) is now proven prime by Paul Underwood [ March 21, 2022 ],
see https://www.mersenneforum.org/showpost.php?p=602219&postcount=35

[ November 23, 2022 ]
Xinyao Chen (email) added the only near-miss circular prime of 12 digits.
and provided me with additional info and an OEIS sequence about near-misses whereby digits 2, 4, 5, 6 and 8 are also allowed.
Also passes through that no near-miss's exist for 10 & 11 and for 13 up to 16 digits.

Two tables have been added to the page from Chen's hand.
The first is about circular primes in bases b ⩽ 12 and the second is about permutable primes in bases b ⩽ 36.

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