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. It is possible that more near misses exists when the digits 2, 4, 5, 6 and 8 are allowed. ( Carlos Rivera examined this case more in 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.
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 rapidlyapproaches 0 as d increases.
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 rapidlyapproaches 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."
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."
Circular prime from "The Prime Glossary".
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
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
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
757 eligible primes
249 eligible primes
63 eligible primes
30 eligible primes
10 eligible primes
3 eligible primes 4 if 1 is taken as a prime ( apart from 2 and 5 )
[ July 11, 2008 ]Justin Chan (email) makes an important contribution ! - go to topic
[ April 11, 2002 ]Walter Schneider (email) improved his search program for circular primes considerably and used it to search for circular primes ofdigit length 20 and 21. No solutions were found.Running time on a Pentium-3 550Mhz was 19 hours for 20 digitsand a couple of days for 21 digits.
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.
Darren Smith (email) from Milwaukee, Wisconsin determined there are no circular primes for lengths 13, 14, 15 and 16.
David W. Wilson (email) determined there are no circular primes for lengths 10, 11 and 12.
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