Palindromes

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pal•in•drome   

Pronunciation: (pal'in-drOm"), [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!

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2002

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

Some number palindromes

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

Palindrome primes

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

The Ubiquitous 196

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

Palindrome Definitions

Some general facts and definitions about palindrome numbers.

Palindrome magic squares

Magic squares constructed with palindromes or a palindrome magic sum.

Word palindromes

Some of my favorites.

2002

The year 2002 is a palindrome. So was the year 1991. Ho-Hum 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 web-site on February 20th.

The sum of an order 11 normal magic square is 671. If you add 112  to each number in the square, the new magic sum is 2002.

Or to say it a different way:     Add 1331 (or 113 ) to 671 gives the new magic sum of S11 = 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
= 12
= 22
= 32
= 42
= 52
= 62
= 72
= 82
= 92

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

100012 = 100020001
110112 = 121242121
111112 = 123454321
112112 = 125686521

111 = 11
11
2
= 121
11
3
= 1331
11
4
= 14641

Cube root not palindromic

22013 = 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

T47 (above)            539593131395935         consists only of the odd digits   1, 3, 5, 9
T60 (above)            8208268228678028       consists only of the even digits  0, 2, 6, 8

                                                                   2664444662

T70 (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 T34, T52,T98, T130. (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 57th positive number palindrome is 484   (57 = Heinz’s number)
The 23456788th positive number palindrome is 12345678987654321

See Eric Schmidt's    http://eric-schmidt.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 9-Digit 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 26-1, pp38-41,  by Harvey Dubner

An Order-3 Superperfect Prime Square

1 of the 88 possible order-3 perfect prime squares (not counting rotations and reflections. Each row, column, and the two main diagonals all consist of 3-digit 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

 

3 1 3
1 5 1
3 1 3

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.

A Palindromic Sequence Series

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 27-digit  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 3-digit 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.

Definitions

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 Reversal-Addition 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 Reversal-Addition 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.

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PALINDROME CHART

Sample of a class worksheet, showing the number of iterations required for each 2-digit number to become a palindrome,

 

 

 

 

 

 

From  Blackstock JHS, Oxnard, California, USA

 

Palindrome magic squares

ares

 
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.
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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 order-8 magic square is a rearrangement of the same 64 unique 3 digit palindromes as the one to the left. Now we have an order-8 pandiagonal magic square where each quarter is an order-4 magic square which is also pandiagonal.  S8 = 4884, S4 = 2442.

Allen Johnson, JRM 21:2, pp 155-156

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Sophie Germaine related palindrome magic squares

 

By Jaime Ayala. See Carlos Rivera's http://www.primepuzzles.net/puzzles/puzz_124.htm

252 171 363
373 262 151
161 353 272


2n + 1

505 343 727
747 525 303
323 707 545

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 canal--Panama!
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)

This page was originally posted February 2002
It was last updated October 22, 2010
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz