Palindromic Prime Statistics  The Table
Palindromic Primes 
LENGTH 
TOTAL NUMBER A040025 
SMALLEST PALINDROMIC PRIME A028989 
LARGEST PALINDROMIC PRIME A028990 
 


10001  + Daniel Heuer  – Jens Kruse Andersen 
10^{10000} + 222999222*10^{4996} + 1 
10^{10001} – 6192916*10^{4997} – 1 
1001  + Harvey Dubner  + D. Heuer  – J. K. Andersen 
10^{1000} + 81918*10^{498} + 1 
10^{1001} – 23332*10^{498} – 1 
101  + Daniel Heuer  – Jens Kruse Andersen 
10^{100} + 303*10^{49} + 1 
10^{101} – 21412*10^{48} – 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  unknown 
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. )
"I'm surprised to see that the successive number of digits in the total of palprimes
from 1digit to 17digits seems to increase by approximately a constant multiplier of 10.
E.g. 45 is ~ 10*4 = 40 (45 actual), 465 is ~ 10*45 = 450 (465 actual), and so on [Sloane A039657].
Therefore, it is easy to guess at the approximate number of 19digit and 21digit palprimes
(I will not hazard a guess beyond that!). Simply divide the estimated total number of digits by the number of digits; thus 4597688420/19 = ~ 241983600 19digit palprimes.
The same procedure yields an estimate of ~ 2189370676 21digit palprimes." 
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 cooperation 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 :
10301 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
Featured in
Prime Curios! 9372 = 10301  929
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.
1092901  1093901
1177711  1178711
1242421  1243421
1280821  1281821
1286821  1287821
1327231  1328231
1362631  1363631
1411141  1412141
1463641  1464641
1489841  1490941
1550551  1551551
1556551  1557551
1579751  1580851
1597951  1598951
1657561  1658561
1684861  1685861
1823281  1824281
1831381  1832381
1878781  1879781  1880881  1881881
1883881  1884881
1908091  1909091
1951591  1952591
1957591  1958591
1968691  1969691  1970791
1981891  1982891
1987891  1988891
3001003  3002003
3064603  3065603
3072703  3073703
3211123  3212123
3222223  3223223
3285823  3286823
3304033  3305033
3364633  3365633
3391933  3392933
3424243  3425243
3443443  3444443
3589853  3590953  3591953
3708073  3709073
3716173  3717173
3721273  3722273
3762673  3763673
3768673  3769673
3773773  3774773
3792973  3793973
3863683  3864683
3997993  3998993

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

“ Every palindrome with an even number of digits is composite  Proof ”
( 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 !
The quickest way perhaps to show that no palindrome except 11 can be a prime if it has
an even number of digits is described in Martin Gardner book “Puzzles from Other Worlds” :
“... for divisibility by 11 is to add all the digits in even positions, then add all the digits in odd
positions. If and only if the difference between these two sums is 0 or a multiple of 11, the number
will be divisible by 11. When a palindrome has an even number of digits, those in the even positions
will duplicate those in odd positions. The two sums will be the same, and their difference will be zero;
hence the number will be a multiple of 11 and therefore composite (not prime).”
“ Almost all palindromes are composite ”
by William D. Banks, Derrick N. Hart and Mayumi Sakata
Source : https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2004/0011/0006/MRL200400110006a010.pdf
Various Sources
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.
Contributions
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 17digit palindromes on [ June 16, 1998 ].
that are probable prime to base2 (2PRP) 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 10001digit 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 19digit and 21digit palprimes some day but I have other
projects now.
https://listserv.nodak.edu/cgibin/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_0_{8}252_0_{8}_1 which should be corrected into 1_0_{8}212_0_{8}_1.
Thanks Farideh for spotting this error. Much appreciated.
[ February 11, 2006 ]
Shyam Sunder Gupta (email) announced the number of 19digit palindromic primes (Message 1351).
He used strongpseudoprime test to 18 bases which is more than sufficient for 19digit numbers.
Also there is not even a single strongpseudoprime(base2) 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 21digit 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 23digit 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
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