More about Circular Primes

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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 :

Repunits Prime Factors

Repunit - E.W. Weisstein

REP-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

%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 1^st prime repunit.

(19 ones) 1111111111111111111 is a Repunit prime.

1111111111111111111

(23 ones) 11111111111111111111111 is a Repunit prime.

11111111111111111111111

317 - Number of digits in the 3^rd 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 ]
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 ]
Length 12 Total = 0	Nihil !	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 2

none exist

Base 3

2, 12, 21

Base 4

13, 31, 113, 131, 311, 11333, 13331, 31133, 33113, 33311

Base 5

12, 21, 23, 32, 34, 43, 1132, 1321, 1424, 2113, 2414, 3211, 4142, 4241, 13234, 14444, 23413, 32341, 34132, 41323, 41444, 44144, 44414, 44441

Base 6

15, 51, 155, 515, 551

Base 7

2, 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 8

2, 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 9

2, 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 10

2, 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 12

2, 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 2	none exist
Base 3	2, 12, 21
Base 4	2, 3, 13, 31, 113, 131, 311
Base 5	2, 3, 12, 21, 23, 32, 34, 43, 14444, 41444, 44144, 44414, 44441
Base 6	2, 3, 5, 15, 51, 155, 515, 551
Base 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, ...
Base 8	2, 3, 5, 7, 15, 35, 37, 51, 53, 57, 73, 75, 3337, 3373, 3733, 7333, ...
Base 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, ...
Base 10	2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 199, 311, 337, 373, 733, 919, 991, ...
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, 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 12	2, 3, 5, 7, B, 15, 51, 57, 5B, 75, B5, 117, 11B, 171, 1B1, 711, B11, 555B, 55B5, 5B55, B555, ...
Base 13	2, 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 14	2, 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 15	2, 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 16	2, 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 17	2, 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 18	2, 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 19	2, 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 20	2, 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 21	2, 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 22	2, 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 23	2, 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 24	2, 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 25	2, 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 26	2, 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 27	2, 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 28	2, 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 29	2, 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 30	2, 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 31	2, 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 32	2, 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 33	2, 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 34	2, 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 35	2, 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 36	2, 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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Patrick De Geest - Belgium

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com