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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 1010000 + 222999222*104996 + 1 10100016192916*104997 – 1
1001+ Harvey Dubner | + D. Heuer | – J. K. Andersen 101000 + 81918*10498 + 1 10100123332*10498 – 1
101+ Daniel Heuer | – Jens Kruse Andersen 10100 + 303*1049 + 1 1010121412*1048 – 1

55unknown 1 + [0]26 + 5 + [0]26 + 1 [9]26 + 313 + [9]26
53unknown 1 + [0]24 + 474 + [0]24 + 1 [9]26 + 8 + [9]26
51unknown 1 + [0]23 + 252 + [0]23 + 1 [9]24 + 181 + [9]24
49unknown 1 + [0]23 + 6 + [0]23 + 1 [9]23 + 050 + [9]23
47unknown 1 + [0]21 + 282 + [0]21 + 1 [9]22 + 787 + [9]22
45unknown 1 + [0]20 + 333 + [0]20 + 1 [9]22 + 4 + [9]22
43unknown 1 + [0]19 + 242 + [0]19 + 1 [9]19 + 88288 + [9]19
41unknown 1 + [0]18 + 161 + [0]18 + 1 [9]19 + 686 + [9]19
39unknown 1 + [0]17 + 737 + [0]17 + 1 [9]18 + 878 + [9]18
37unknown 1 + [0]15 + 10901 + [0]15 + 1 [9]17 + 868 + [9]17
35unknown 1 + [0]16 + 8 + [0]16 + 1 [9]16 + 848 + [9]16
33unknown 1 + [0]15 + 3 + [0]15 + 1 [9]15 + 838 + [9]15
31unknown 1 + [0]13 + 242 + [0]13 + 1 [9]14 + 626 + [9]14
29unknown 1 + [0]13 + 3 + [0]13 + 1 [9]14 + 4 + [9]14
27unknown 1 + [0]12 + 8 + [0]12 + 1 [9]13 + 1 + [9]13
25unknown 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 1-digit to 17-digits 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 19-digit and 21-digit 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 19-digit palprimes.
The same procedure yields an estimate of ~ 2189370676 21-digit 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 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
153036643 100000 323 000001 999999 787 999999
13353701 100000 8 000001 99999 878 99999
1142042 10000 5 00001 99999 1 99999
95172 1000 3 0001 999 727 999
7668 100 3 001 99 898 99
593 10 3 01 9 868 9
315 1 0 1 9 2 9
21 11
The only palindromic prime
with even number of digits !
11
14
( 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

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)."

Various Sources

 Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online : Neil Sloane's Integer Sequences The palindromic primes statistics are categorised as follows: %N Number of prime palindromes with n digits. under A016115 %N Number of prime palindromes with 2n+1 digits. under A040025 %N Smallest palindromic prime with 2n+1 digits. under A028989 %N Largest palindromic prime with 2n+1 digits. under A028990 Near exhaustive list of all the "palindromic primes" sequences Click here to view some of the author's [P. De Geest] entries to the table. 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 :

 1.32398... 7 11 101 131 161 181 191 313 353 373 383 727 757 3883 5172 9372 10301 11311 12421 13331 16361 16661 21712 30103 32423 34543 69696 71317 76667 78787 81918 94649 97579 98689 101101 333667 606606 969969 1212121 1777771 1934063 3036643 7913197 9230329 27045226 53205635 103323301 188888881 345676543 363818363 906343609 968666869 14432823441 123455554321 1666666666661 151515151515151 999998727899999 74747474747474747 1023456987896543201 1234567894987654321 11111111111111111111111 18133...33181 (35-digits) 19191...19191 (39-digits) 31415...51413 (53-digits) 10080...08001 (175-digits) 91111...11119 (247-digits) 31300...00313 (313-digits)

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 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

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