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Circular Primes  
Home Primes Palindromic Primes Undulating Primes Palindromic Wing Primes Repunits and their prime factors 
1193 is prime 1931 is prime 9311 is prime 3119 is prime 1193 back to starting position 
It is immediate clear that a circular prime only consists of the digits 1, 3, 7 and 9.
The singledigit 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 socalled 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 ] 
http://www.lacim.uqam.ca/~plouffe/OEIS/archive_in_pdf/Absolute_Primes.pdf Proves (actually resummarizes) that a ndigit permutable prime (excluding repunits) does not exist 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)
ddigit primes (this estimate is on the high side, since it includes primes with fewer than d digits).
The probability that a ddigit number is prime is therefore about 1/log(10^d).A ddigit 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 ddigit 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 0.00023869488522460123This argument is obviously very imprecise, but I think the general conclusion is valid,
namely that the expected number of ddigit nonrepunit 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
As you can see there is nothing very special about my search program. One of the crucial facts
make a fermat test.
I think is the fast generation using strings not numbers and the quick test in step 3."
Click here to view some entries to the table about palindromes. 
Circular prime from “The Prime Glossary”.
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 (10^10311)/9 
Length 317 Total = ?  R317  Repunit 317 (10^3171)/9 
Length 25 Total = ? 
Unknown.  ? 
Length 24 Total = ? 
Unknown.  ? 
Length 23 Total = 1  R23  11111111111111111111111 is the only one !

Length 22 Total = 0 
Nihil ! 

Length 21 Total = 0 
Nihil ! 

Length 20 Total = 0 
Nihil ! 

Length 19 Total = 1  R19  1111111111111111111 is the only one !

Length 18 Total = 0 
Nihil ! 

Length 17 Total = 0 
Nihil ! 

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 ! 

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 ! 

Length 10 Total = 0 
Nihil ! 

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 ) 
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 nonperfect 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  
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 nonperfect 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 nearrepdigit, 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). 
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 PentiumIII 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 Pentium3 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
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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