pal•in•drome
Pronunciation: (pal'indrOm"),
[key] —n.
a word, line, verse, number, sentence, etc., reading the
same backward as forward,
as Madam, I'm Adam or
Poor Dan is in a droop. (
http://www.infoplease.com/ ) 
There are probably over 100 pages on the World Wide Web
with palindromes as their primary subject. However, most of them deal with word,
sentence, poem, etc. palindromes.
Because these pages are on the subject of number patterns,
the emphasis will be on numbers ( and some equations) that read the same
backward as forward. All these number palindromes will be in our decimal number
system, although of course, palindromes exist in all number systems. I will
include links to many related Web sites for those who wish to
pursue the matter further. For the sake of completeness, I will include a small
section of my favorite word palindromes.
The inspiration for a page on this subject is the fact that
2002 (the year this was
written) is a palindromic year!

On the palindromic year 2002. Also, the
Universal Day of Symmetry. 

Numbers and patterns of numbers that read the
same forward and backward. 

Called palprimes, these are
prime numbers that are also palindromic. 

Any (?) number will become a palindrome if you
reverse the digits and add, etc. 

Some general facts and definitions about
palindrome numbers. 

Magic squares constructed with palindromes or a
palindrome magic sum. 

Some of my favorites. 
2002
The year 2002 is
a palindrome. So was the year 1991.
HoHum you say?
Did you realize the previous occurrence of two
palindromic years in one person’s lifetime was the years
999 and
1001!
The next such pair will be 2992
and 3003. So this
happens about once every 1000 years.
The next normal palindromic year will be 2112.
Universal Day of Symmetry
8:02 P.M. on Feb. 20, 2002 is a very unique time and date.
It may be written as 20:02, 20/02, 2002 (Canadian, South
American, & European date format, 24 hour clock system).
Without the punctuation it becomes:
2002
2002 2002
which is three palindromic identical numbers in a row. It is also a
palindromic sentence because the three numbers taken together are the same when
reversed. A similar symmetric date and time happened was some years ago.
10:01 AM, on January 10, 1001, to be exact. write it
1001 1001 1001
Because the clock can only go to 23:59, this situation can never again be
repeated.
Addendum (Feb.20):
Jacques Misguich (France) pointed out 11:11 of Nov. 11, 1111
write it 1111 1111 1111
Carlos Rivera (Mexico) pointed out 21:12 of Dec. 21, 2112
write it 2112 2112 2112
Addendum2 (Feb.21):
The above 3 examples are written using the Canadian date format (as
mentioned at the start).
Aale de Winkel (The Netherlands) pointed out that they may also
be written using the U.S.A. numeric
date format which is mm/dd. So Rivera's example then would be written
2112 1221 2112
And another date would be 12:21, Dec. 21, 1221, or
1221 1221
1221
Ivan Skvarca (web
site has disappeared) has proposed: 
In order to celebrate such a remarkable event, we'll
celebrate the
Universal Day of Symmetry on February 20th.
We will celebrate it by exchanging materials, articles, creations and
findings related to the ludic and recreational aspects of symmetry, which
will be published simultaneously in this website on February 20th. 
The sum of an
order
11
normal magic square is 671. If you add
11^{2
} to each number in the square, the new magic sum
is
2002.
Or to say it a
different way: Add
1331
(or 11^{3}
) to 671 gives the new magic sum of S_{11}
= 2002.
Suggested by
Aale de Winkel
There are many Web sites dealing with the number properties of
2002.
One of the best is Patrick De Geest's
http://www.worldofnumbers.com/ He also has an extensive list of links
to other Palindrome pages.
Some number
palindromes
1
1+2+1
1+2+3+2+1
1+2+3+4+3+2+1
1+2+3+4+5+4+3+2+1
1+2+3+4+5+6+5+4+3+2+1
1+2+3+4+5+6+7+6+5+4+3+2+1
1+2+3+4+5+6+7+8+7+6+5+4+3+2+1
1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1 
=
1
= 2+2
= 3+3+3
= 4+4+4+4
= 5+5+5+5+5
= 6+6+6+6+6+6
= 7+7+7+7+7+7+7
= 8+8+8+8+8+8+8+8
= 9+9+9+9+9+9+9+9+9 
=
1^{2}
= 2^{2}
= 3^{2}
= 4^{2}
= 5^{2}
= 6^{2}
= 7^{2}
= 8^{2}
= 9^{2} 
Palindromic Triangular
n n(n+1)/2
11 66
1111 617716
111111 6172882716
These are 3 of the 40 palindromic
triangular numbers with n < 10,000,000.
Unfortunately, the next number in the above series (11111111) is not palindromic,
although it does contain all 10 digits.
JRM 6:2,p146 &8:2, p92
Palindromic
Squares of Palindromes
10001^{2} = 100020001
11011^{2} = 121242121
11111^{2} = 123454321
11211^{2} = 125686521
11^{1}
=
11
11^{2}
= 121
11^{3}
= 1331
11^{4}
= 14641
Cube root not palindromic
2201^{3} = 10662526601
The only known palindromic cube
whose root is not palindromic !
From Square.htm by Patrick
De Geese, Belgium
Palandromic powers pattern
Base 
Square 
Cube 
Forth power 
11 
121 
1331 
14641 
101 
10201 
1030301 
104060401 
1001 
1002001 
1003003001 
1004006004001 
And so on ... 
In each case, the groups of zeros inserted is equal to the
group of zeros in the base.
The number of groups (of zeros) is equal to the power.
8 x 8 + 13
= 77
88 x 8 + 13
= 717
888
x 8 + 13 =
7117
8888 x 8 + 13
=
71117
88888 x 8 + 13
= 711117
And so on...
Interesting
Palindromic Triangular Numbers
539593131395935
8208268228628028
T_{47} (above)
539593131395935
consists only of the odd digits 1,
3, 5, 9
T_{60} (above)
8208268228678028
consists only of the even digits 0, 2, 6, 8
2664444662
T_{70}
(above)
2664444662 = 2 x 11 x 121111121
Three prime palindromic factors !
6677
191
7766
6677
446448
7766
6677 12035788 60
88753021
7766
6677 12035788 7130286820317 88753021 7766
Index numbers above are T_{34},
T_{52},T_{98}, T_{130}. (Index numbers indicate the rank
of the palindromic number.
Spaces are for illustration only.
All of the above from
Triangle.htm by Patrick De Geese, Belgium (June 1996)
Multiples of 9 (with a 9 at the end)
918273645546372819
Products of consecutive numbers
77x78 = 6006
77x78x79 = 474474
Fascinating Palindromes
Start
palindrome 
Divided by 
Gives this
palindrome 
Divided by 
Gives this
palindrome 
121 
11 
11 
11 
1 
1234321 
11 
112211 
11 
10201 
12345654321 
11 
1122332211 
11 
102030201 
123456787654321 
11 
11223344332211 
11 
1020304030201 
This pattern constructed from material on Peter Collins
http://www.iol.ie/~peter/num1.html
He calls these palindromes 'Fascinating'!
The 57^{th} positive number
palindrome is 484 (57 =
Heinz’s number)
The 23456788^{th} positive number palindrome is
12345678987654321
See Eric Schmidt's
http://ericschmidt.com/eric/palindrome/index.html to find the ranking of
your favorite palindrome.
Palindrome prime number
patterns
Depression Primes
727
757
72227 75557
722222227
75555555557
The above numbers are called
depression primes. The next ones in the 'two' series contain 27 and 63 two's!
Note the 'seven' two's in the one above. The next ones in the 'five' series
contain 19, 21, 57, 73 & 81 fives.
JRM 25:1 p51 by Chris
Caldwell
Interesting 9Digit
Palindromic Primes
188888881 111181111 323232323
199999991 111191111 727272727
355555553 777767777 919191919
Plateau Primes
8 like digits
Smoothly Undulating
123494321
765404567
345676543 354767453 987101789
345686543 759686957 987646789
Peak Primes
5 consecutive digits
Valley Primes
Palindromic Primes
There are a total of 5172 nine digit
primes that read the same forward or backward. Many of them have extra
properties.
Plateau Primes
There are 3 primes where all the interior digits are alike and are
higher then the terminal digits. There are two primes ,
322222223
& 722222227
in which the interior digits are smaller then the end ones. These are called
Depression Primes
Undulating Primes
So called when adjacent digits are alternately greater or less then their
neighbors. If there are only two distinct digits, they are called smoothly
undulating. Of the total of 1006 undulating nine digit palindromic primes, seven
are smoothly undulating.
Peak & Valley Primes
If
the digits of the prime, reading left to right, steadily increase to a maximum
value, and then steadily decrease, they are called peak primes.
Valley primes are just the opposite. There are a total of 10 peak and 20
valley primes.
345676543
is unique because of the five consecutive digits.
JRM 14:1 p30
Palindromic Sophie
Germain Primes
191 & 383
39493939493 & 78987878987
If P is greater then 2 and is a
prime, then if 2P+1 is also prime, P is known as a Sophie Germain prime. There
are many such primes but only 71 such pairs with three to eleven digits if both
primes are palindromic. Above are the lowest and the highest such pairs.
If Q (2P+1) is itself a Sophie
Germain prime, a total of 19 such triplets where each prime is palindromic have
been discovered. They range in size from 23 digits long to 39 digits long
The smallest such triplet follows:
19091918181818181919091
38183836363636363838183 76367672727272727676367
JRM 261, pp3841, by Harvey Dubner
An
Order3 Superperfect Prime Square
1 of the 88 possible
order3 perfect prime squares (not counting rotations and
reflections. Each row, column, and the two main diagonals all
consist of 3digit primes when read in either direction. This one is
superperfect because the partial diagonal pairs are also
prime numbers. The 5 can be replaced with an 8


Repunit
Primes
1
1111111111111111111
(19 ones)
11111111111111111111111
(23 ones)
11111111111..........11111111111
(317 ones)
11111111111..................11111111111 (1031
ones)
These are all the
prime numbers smaller then 10,000 ones that contain the digit 1 only. Numbers
that contain the digit 1 only ( in the decimal system) are known as repunits.
They can only be prime if the number of 1's is prime. Repunits, because they
read the same backward as foreword, are palindromes, although, admittedly, not
all that interesting.
A Special Palindromic
Prime
1888081808881
This
number reads the same upside down or when viewed in a
mirror.
Each sequence is formed from the one above it by inserting n,
the row number, between all adjacent numbers that add to n. k
is the number of numbers in each sequence. So far all k are
prime numbers. Does this series continue indefinitely?
This pattern is credited to Leo Moser (Martin Gardner, The Last
Recreations, p.199).
See 12 palprime patterns already on my
Primes page
and 3 more patterns on my
moreprimes
page
As an example of palindromic primes, here is a pyramid (list) of
palindromic primes supplied by G. L. Honaker, Jr.
2
30203
133020331
1713302033171
12171330203317121
151217133020331712151
1815121713302033171215181
16181512171330203317121518161
331618151217133020331712151816133
9333161815121713302033171215181613339
11933316181512171330203317121518161333911
Chris Caldwell and G.
L. Honaker, Jr.’s
http://primes.utm.edu/glossary/page.php/PalindromicPrime.html
Ten 27digit palindromic primes in
arithmetic progression
Found April 23, 1999
About 20 PC's were used, with the search team consisting of: Harvey
Dubner, Manfred Toplic, Tony Forbes, Jonathan
Johnson, Brian Schroeder and Carlos
Rivera.
742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247
742950290879090978092059247
Common difference = 1010100000000000 (divisible by
2,3,5,7)
http://listserv.nodak.edu/scripts/wa.exe?A2=ind9904&L=nmbrthry&F=&S=&P=1602
347182965 /(3+4+7+1+8+2+9+6+5)=
7715177 (prime!)
Carlos River’s Prime Puzzles & Problems
http://www.primepuzzles.net/puzzles/puzz_041.htm
8009 is the first prime p for which the decimal period of 1/p is
2002.
1878781
1880881
Two related palprime pairs
1879781 1881881
Patrick De Geest – Belgium
http://www.worldofnumbers.com/index.html
Patrick has much material on this subject, and many links to other palindromic
sites.
The smallest palindromic prime containing all 10 digits is
1023456987896543201.
Ivar Peterson’s MathTrek
at
http://www.sciencenews.org/sn_arc99/5_8_99/mathland.htm
The Ubiquitous 196
Take any positive integer of two digits or more,
reverse the digits, and add to the original number.
If the resulting number is not a palindrome, repeat the procedure with
the sum until
the resulting number is a palindrome.
For example, start with
87 or 88 or 89. Applying this process, we obtain: 
87 
88 
89 
87 + 78 = 165 
88
is a palindrome 
89 + 98 = 187 
165 + 561 = 726 

187 + 781 = 968 
726 + 627 = 1353 

968 + 869 = 1837 
1353 + 3531 = 4884 

until finally after 24 steps 
4884
is a palindrome 

becomes
8813200023188 
Using the above algorithm, Do all numbers
become palindromes eventually? The answer to this problem is not known.
The venerable David Wells (Curious and Interesting Numbers, pp.211, 212)
says that “196 is the only number less then 10,000 that by this
process has not yet produced a palindrome.”
Many palindrome web sites imply the same thing, but a little reflection reveals
that is not correct. What is correct is that 196 is the
smallest number that may not produce a palindrome.
Of the 900 3digit numbers 90 are palindromic and 735 require from 1 to 5
reversals and additions.
Of the remaining 75 numbers, most form chains of numbers that eventually
result in a palindrome. One such chain (part of the 89 chain in the example
above) is 187, 286, 385, 583, 682, 781,869, 880, 968.
Others (of the 75) form a chain that so far has not resulted in a palindrome.
This chain starts with the 196 mentioned above, The first few
numbers of this chain are 196, 887, 1675, 7436, 13783. Each number in this chain
must also be included with the 196 as possibly not converging to a palindrome.
Mathematicians are unable to prove that these numbers will
eventually form a palindrome.
Consequently, a tremendous amount of time and effort has been expended in the
search to resolve this issue.
By Sept. 11, 2003, Wade VanLandingham (Florida, U.S.A.) had
tested the number 196 to 278,837,830 iterations, resulting in a number of
117,905,317 digits. It was still not a palindrome! Wade has tested a number of
programs written by different people to perform this search. The one he is
currently using was written by Eric Goldstein of the Netherlands, and is the
fastest to date.
LYCHREL NUMBERS: ALL numbers
that do not form a palindrome through the reverse and add process.
Examples:
196 (Seed), 295(Kin), 394(Kin), 879 (Seed), 887(Kin), 1997(Seed)...
SEED NUMBERS: A subset of
Lychrel Numbers, that is the smallest number of each non palindrome
producing thread. A Seed number may be a palindrome itself.
Examples:
196, 879, 1997... 9999, 99999, 999999
KIN NUMBERS: A subset of
Lychrel Numbers, that include all numbers of a thread, except the Seed, or
any number that will converge on a given thread after a single +iteration.
This term was introduced by Koji Yamashita in 1997.
Examples:
295, 394, 493, 978, 2996... 
Wade VanLandingham has a large site on
Lychrel numbers.
His pages contain a lot of information and many links to other sites concerned
with the 196 problem.
By March 31, 2003, he had found 4,455,557 Seed numbers
among the 14 digit numbers. On October 27, 2003 Wade told me that all seed
numbers (in the 14 digit set) have been found.
Jason Doucette is tackling the 196 problem
from a different angle. He is searching each range of x digit integers for
the number that requires the largest number of iterations to become a
palindrome.
His newest record is the 19 digit number
1,186,060,307,891,929,990 which takes 261 iterations to resolve into a 119 digit
palindrome.
The following table is supplied courtesy of Jason Doucette
Common ReversalAddition Tests
(Most Delayed Palindromic Number record holders for each digit
length) 
Digit Length 
Number 
Result 
2 
89 
solves in 24 iterations. 
3 
187 
solves in 23 iterations. 
4 
1,297 
solves in 21 iterations. 
5 
10,911 
solves in 55 iterations. 
6 
150,296 
solves in 64 iterations. 
7 
9,008,299 
solves in 96 iterations. 
8 
10,309,988 
solves in 95 iterations. 
9 
140,669,390 
solves in 98 iterations. 
10 
1,005,499,526 
solves in 109 iterations. 
11 
10,087,799,570 
solves in 149 iterations. 
12 
100,001,987,765 
solves in 143 iterations. 
13 
1,600,005,969,190 
solves in 188 iterations. 
14 
14,104,229,999,995 
solves in 182 iterations. 
15 
100,120,849,299,260 
solves in 201 iterations. 
16 
1,030,020,097,997,900 
solves in 197 iterations. 
17 
10,442,000,392,399,960 
solves in 236 iterations. 
18 
170,500,000,303,619,996 
solves in 228 iterations. 
19 
1,186,060,307,891,929,990 
World
Record  solves in 261 iterations. 
See more on the
ReversalAddition
Palindrome Test on 1,186,060,307,891,929,990.
Or see Jason’s site on palindromes and the 196 problem at
http://www.jasondoucette.com/worldrecords.html
ADDENDUM: Nov. 30, 2005
I have added 2 lines to the above table and changed several sentences, in
response to an email from Jason Doucette.
PALINDROME CHART Sample of
a class worksheet, showing the number of iterations required for each
2digit number to become a palindrome,
From Blackstock JHS, Oxnard,
California, USA


Palindrome magic
squares
494 
501 
500 
505 
508 
497 
502 
495 
501 
496 
507 
498 
499 
506 
493 
504 

388 
394 
401 
407 
412 
402 
405 
410 
389 
396 
411 
391 
397 
400 
403 
395 
398 
404 
413 
392 
406 
414 
390 
393 
399 

Two pandiagonal magic squares with the magic sum of
2002.
363 
424 
646 
747 
757 
767 
787 
393 
696 
232 
383 
898 
939 
969 
242 
525 
676 
949 
222 
595 
737 
888 
272 
545 
656 
868 
959 
666 
444 
373 
353 
565 
636 
343 
484 
333 
999 
626 
878 
585 
535 
292 
777 
848 
262 
555 
929 
686 
494 
979 
838 
323 
282 
252 
989 
727 
828 
797 
575 
474 
464 
454 
434 
858 

424 
777 
353 
888 
878 
323 
787 
454 
393 
848 
464 
737 
747 
494 
838 
363 
868 
333 
797 
444 
434 
767 
343 
898 
757 
484 
828 
373 
383 
858 
474 
727 
525 
999 
232 
686 
979 
545 
666 
252 
292 
626 
585 
939 
646 
272 
959 
565 
989 
535 
696 
222 
555 
969 
242 
676 
636 
282 
929 
595 
262 
656 
575 
949 

This is
an order 4 pandiagonal magic square consisting of all palindromic
numbers and has the magic palindromic sum of 2442. It is bordered
to make an order 6 square with the magic sum 3663, and an order 8
square with the magic sum 4884, both of which are also palindromic.
Note the main diagonals of the order 4 square consists of the
repdigits 222
to 999. 
This
order8 magic square is a rearrangement of the same 64 unique 3
digit palindromes as the one to the left. Now we have an order8
pandiagonal magic square where each quarter is an order4 magic
square which is also pandiagonal.
S_{8} = 4884, S_{4} = 2442.
Allen Johnson, JRM 21:2, pp
155156 
Word palindromes
RADAR (the most popular word)
Ogopogo (the mythical inhabitant of
Okanogan Lake, BC, Canada)
Glenelg (ON, Canada)
Kanakanak (AK, USA)
Wassamassaw
(SC, USA)
A Dan, a clan, a canal  Canada!
A man, a plan, a cat, a ham, a yak, a yam, a hat, a canalPanama!
Go hang a salami, I'm a
lasagna hog
I saw desserts; I'd no
lemons, alas no melon. Distressed was I.
Madam, I'm Adam (Probably the most
popular palindrome phrase)
Poor Dan is in a droop
Too far, Edna, we wander afoot.
Feb. 20, 2002 (the day this page was
posted) marks the 34th anniversary of the ABC television show "Columbo,"
which inspired this palindrome, written by a University of Minnesota
mathematician:
Murdered  no pistol, so no grenade,
Dan. Ergo no slots. I pondered: Rum!
http://www1.keenesentinel.com/localnews/story2.htm (Feb.20/02)
