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 Palindromic Primes 1 Page 1 | |||
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| Palindromic Primes | |||
|---|---|---|---|
| LENGTH | TOTAL NUMBER A040025 |
SMALLEST PALINDROMIC PRIME A028989 |
LARGEST PALINDROMIC PRIME A028990 |
| 10001 | + Daniel Heuer | – Jens Kruse Andersen | 1010000 + 222999222*104996 + 1 | 1010001 – 6192916*104997 – 1 |
| 1001 | + Harvey Dubner | + D. Heuer | – J. K. Andersen | 101000 + 81918*10498 + 1 | 101001 – 23332*10498 – 1 |
| 101 | + Daniel Heuer | – Jens Kruse Andersen | 10100 + 303*1049 + 1 | 10101 – 21412*1048 – 1 |
| 55 | unknown | 1 + [0]26 + 5 + [0]26 + 1 | [9]26 + 313 + [9]26 |
| 53 | unknown | 1 + [0]24 + 474 + [0]24 + 1 | [9]26 + 8 + [9]26 |
| 51 | unknown | 1 + [0]23 + 252 + [0]23 + 1 | [9]24 + 181 + [9]24 |
| 49 | unknown | 1 + [0]23 + 6 + [0]23 + 1 | [9]23 + 050 + [9]23 |
| 47 | unknown | 1 + [0]21 + 282 + [0]21 + 1 | [9]22 + 787 + [9]22 |
| 45 | unknown | 1 + [0]20 + 333 + [0]20 + 1 | [9]22 + 4 + [9]22 |
| 43 | unknown | 1 + [0]19 + 242 + [0]19 + 1 | [9]19 + 88288 + [9]19 |
| 41 | unknown | 1 + [0]18 + 161 + [0]18 + 1 | [9]19 + 686 + [9]19 |
| 39 | unknown | 1 + [0]17 + 737 + [0]17 + 1 | [9]18 + 878 + [9]18 |
| 37 | unknown | 1 + [0]15 + 10901 + [0]15 + 1 | [9]17 + 868 + [9]17 |
| 35 | unknown | 1 + [0]16 + 8 + [0]16 + 1 | [9]16 + 848 + [9]16 |
| 33 | unknown | 1 + [0]15 + 3 + [0]15 + 1 | [9]15 + 838 + [9]15 |
| 31 | unknown | 1 + [0]13 + 242 + [0]13 + 1 | [9]14 + 626 + [9]14 |
| 29 | unknown | 1 + [0]13 + 3 + [0]13 + 1 | [9]14 + 4 + [9]14 |
| 27 | unknown | 1 + [0]12 + 8 + [0]12 + 1 | [9]13 + 1 + [9]13 |
| 25 | Announced in Number Theory List Message on [ December 10, 2014 ] by Shyam Sunder Gupta Type 1...1 = 46259767358 Type 3...3 = 45553664943 Type 7...7 = 44716368322 Type 9...9 = 44285590742 180815391365 |
1 + [0]10 + 161 + [0]10 + 1 | [9]11 + 494 + [9]11 |
| 23 | Announced in Number Theory List Message on [ October 4, 2013 ] by Shyam Sunder Gupta Type 1...1 = 5027672681 Type 3...3 = 4955286148 Type 7...7 = 4892461690 Type 9...9 = 4867380045 19742800564 |
1 + [0]10 + 3 + [0]10 + 1 | [9]10 + 757 + [9]10 |
| Enoch Haga wrote [ February 28, 1999 ] ( Read also about his observation at Carlos Rivera's PP&P website Conjecture 14. ) |
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| 21 | Announced in Number Theory List Message 2104 on [ March 13, 2009 ] by Shyam Sunder Gupta Type 1...1 = 552358972 Type 3...3 = 540945484 Type 7...7 = 533119350 Type 9...9 = 531667127 2158090933 |
1 + [0]8 + 212 + [0]8 + 1 | [9]9 + 757 + [9]9 |
| 19 | Announced in Number Theory List Message 1351 on [ February 6, 2006 ] by Shyam Sunder Gupta Type 1...1 = 61960057 Type 3...3 = 60731724 Type 7...7 = 58734513 Type 9...9 = 57667571 239093865 |
1 + [0]8 + 8 + [0]8 + 1 | [9]9 + 2 + [9]9 |
| 17 | Found by co-operation from Martin Eibl Carlos Rivera Warut Roonguthai Type 1...1 = 6891972 Type 3...3 = 6755263 Type 7...7 = 6698063 Type 9...9 = 6699928 27045226 |
1 + [0]7 + 5 + [0]7 + 1 | [9]8 + 2 + [9]8 |
| 15 | 3036643 | 100000 323 000001 | 999999 787 999999 |
| 13 | 353701 | 100000 8 000001 | 99999 878 99999 |
| 11 | 42042 | 10000 5 00001 | 99999 1 99999 |
| 9 | 5172 | 1000 3 0001 | 999 727 999 |
| 7 | 668 | 100 3 001 | 99 898 99 |
| 5 | 93 | 10 3 01 | 9 868 9 |
| 3 | 15 | 1 0 1 | 9 2 9 |
| 2 | 1 | 11 The only palindromic prime with even number of digits ! | 11 |
| 1 | 4 ( 5 if 1 is counted as a prime...) |
2 1* | 7 |
Here are the 15 palindromic primes of length 3 :
101 131 151 181 191 313 353 373 383 727 757 787 797 919 929
And here are the 93 palindromic primes of length 5 :
Featured in Prime Curios! 9372 = 10301 - 92910301 10501 10601 11311 11411 12421 12721 12821 13331 13831 13931 14341 14741 15451 15551 16061 16361 16561 16661 17471 17971 18181 18481 19391 19891 19991 30103 30203 30403 30703 30803 31013 31513 32323 32423 33533 34543 34843 35053 35153 35353 35753 36263 36563 37273 37573 38083 38183 38783 39293 70207 70507 70607 71317 71917 72227 72727 73037 73237 73637 74047 74747 75557 76367 76667 77377 77477 77977 78487 78787 78887 79397 79697 79997 90709 91019 93139 93239 93739 94049 94349 94649 94849 94949 95959 96269 96469 96769 97379 97579 97879 98389 98689
Enumerating all 668 palindromic primes of length 7 is a task too daunting
so I will confine myself to the subset of the Palindromic Prime "Twin Pairs" of which there are 83 (duo's).
There are three palindromic primes in consecutive rows of three (triples) and one of four (quartet).
Hereunder they are displayed in a darkviolet color.
|
7035307 - 7036307 7114117 - 7115117 7155517 - 7156517 7158517 - 7159517 7249427 - 7250527 7256527 - 7257527 7485847 - 7486847 7507057 - 7508057 7518157 - 7519157 7665667 - 7666667 7668667 - 7669667 7782877 - 7783877 7819187 - 7820287 - 7821287 7831387 - 7832387 7867687 - 7868687 7957597 - 7958597 7984897 - 7985897 9042409 - 9043409 9045409 - 9046409 9109019 - 9110119 9127219 - 9128219 9173719 - 9174719 9199919 - 9200029 9222229 - 9223229 9230329 - 9231329 9439349 - 9440449 9492949 - 9493949 9585859 - 9586859 9601069 - 9602069 9732379 - 9733379 9781879 - 9782879 9787879 - 9788879 9817189 - 9818189 9836389 - 9837389 9888889 - 9889889 9907099 - 9908099 9918199 - 9919199 9926299 - 9927299 9931399 - 9932399 9980899 - 9981899 |
( this proof works in every base )
Except for 11 which is the only existing palindromic prime with an 'even' number of digits.
Every other palindrome with an even number of digits is divisible by 11 and so can't be prime.
For instance 98766789 = 11 x 8978799 !
Have a look at Shareef Bacchus's proof by induction.
On [ August 20, 1997 ] Neo Chee Beng (email) from Singapore mailed me a more elegant proof from the point whereby
Shareef Bacchus showed that an even palindrome is a sum of multiples of 10^(2k+1)+1.
Neo Chee Beng continues by using congruency arithmetic. Here is his alternate proof :
10^(2k+1)+1 = (-1)^(2k+1)+1 (mod 11)
= -1 + 1 (mod 11)
= 0 (mod 11)
which means that 10^(2k+1)+1 is divisible by 11. Indeed, a very short proof !
an even number of digits is described in Martin Gardner book “Puzzles from Other Worlds” :
by William D. Banks, Derrick N. Hart and Mayumi Sakata
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Click here to view some entries to the table about palindromes. |
Prime Curios! - site maintained by G. L. Honaker Jr. and Chris Caldwell
Here are a few palindrome related entries :
Use the Search our Curios! to find more palindrome related entries.
Martin Eibl (email) provided me the total number of palindromic primes of length 11, 13 and 15.
He reached those results using a clever program written in UBASIC.
Three hours before Carlos Rivera, Martin also ended the count for palindromic primes of length 17 on [ June 9, 1998 ].
Carlos Rivera 
from Nuevo León, México
corrected the value of palindromic primes of length 13 from 352701 to 353701.
Carlos and his team finalized the counting of palindromic primes of length 17 on [ June 9, 1998 ].
Unfortunately the results of Martin (27045226) and Carlos (27045217) differ.
Warut Roonguthai (email) from Thailand independently recounted the number of 17-digit palindromes on [ June 16, 1998 ].
that are probable prime to base-2 (2-PRP) by using a selfwritten UBASIC program.
The result of his work is that he strongly supports Martin Eibl's total (same total) !
Carlos rechecked his program... and confirms [ June 25, 1998 ] Mr Eibl's count.
The score is settled now - goto topic
Neo Chee Beng (email) from Singapore embellished the proof that every even palindrome is divisible by 11.
[ February 15 & 17, 2005 ]
Jens Kruse Andersen (email) from Denmark sent in the largest palprimes with 101, 1001 and 10001 digits.
Found and proved with PrimeForm.
" The Prime Pages database says Harvey Dubner found the smallest titanic palprime in 1988: 65702 10^1000+81918*10^498+1 1001 D 1988 Palindrome http://primes.utm.edu/primes/page.php?id=58841
The submission on this page by Daniel Heuer is of later date and was an independant rediscovery.
The smallest gigantic palprime is not in the database.
There is only this one to be found from 1990:24804 10^10002+1232321*10^4998+1 10003 D 1990 Palindrome "
I have written an unreleased palprime sieve PalSieve and once sieved to 10^12
before prp'ing (by Harvey Dubner on May 18, 2004):
PalSieve was used to find the largest 101, 1001 and 10001-digit palprime,
exactly the form it was written for. 101 and 1001 digits is trivial.
It sieved to 10^9 for 10001 digits. For comparison, released PrimeForm
versions factor to 2873654 with pfgw -f. Sieving to 10^9 reduces prp'ing by 28%.
If you want to sieve deep for a large palprime with a fixed number of digits
then let me know.
Maybe I will count 19-digit and 21-digit palprimes some day but I have other
projects now.
https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;aef2870a.0405&S=
[ September 25, 2005 ]
Farideh Firoozbakht (email) pointed out a mistake for the smallest palprime of length 21
in the above table for 1_08252_08_1 which should be corrected into 1_08212_08_1.
Thanks Farideh for spotting this error. Much appreciated.
[ February 11, 2006 ]
Shyam Sunder Gupta (email) announced the number of 19-digit palindromic primes (Message 1351).
He used strongpseudoprime test to 18 bases which is more than sufficient for 19-digit numbers.
Also there is not even a single strongpseudoprime(base-2) known which is palindromic
(see Can You Find [CYF 2] ).
Total time spent was about 80 hours on Pentium IV machine 3.0 GHz.
The results have been checked using UBASIC and Fortran. - goto topic
[ March 13, 2009 ]
Shyam Sunder Gupta (email) announced the number of 21-digit palindromic primes (Message 2104).
" I have announced the number of 21 digit palindromic primes, which I have computed recently.
Total 21 digit palindromic primes are 2158090933." - goto topic
[ October 4, 2013 ]
Shyam Sunder Gupta (email) announced the number of 23-digit palindromic primes (Message).
" Here I now announce the number of 23 digit palindromic primes, which I have computed recently.
Total 23 digit palindromic primes are 19742800564." - goto topic
[ December 19, 2024 ]
Shyam Sunder Gupta (email) announced the number of 25-digit palindromic primes (Message).
" I am happy to report the number of 25-digit palindromic primes, which I have computed about one year back.
Approximately 3600 hours of CPU time (Intel CPU 11700k) was spent on computing all 25-digit palindromic primes.
Total 25 digit palindromic primes are 180815391365." - goto topic
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Patrick De Geest - Belgium
- Short Bio - Some PicturesE-mail address : pdg@worldofnumbers.com