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 Circular Primes | |||
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Introduction
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'.
|
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 ] |
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  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 0.00023869488522460123This 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
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."
Sources Revealed
Here's a reflection of them in a few websites :
Repunits Prime FactorsRepunit - E.W. WeissteinREP-UNIT & REP-DIGIT|
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 (10^1031-1)/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 !
|
| 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 ) |
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 | |
| 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
Message forwarded to me by Xinyao Chen (email).
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 PicturesE-mail address : pdg@worldofnumbers.com