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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
is a palindromic year!
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 statistics |
Some general facts about palindrome numbers. |
Palindrome magic squares |
Magic squares constructed with palindromes or a palindrome magic sum. |
Word palindromes |
Some of my favorites. |
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.
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 |
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.
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 |
This example shows palindromic phrases.
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
203313 x 657624 = 426756 x 313302
The right side of the equal sign is the reverse of the left side.
Palindromic Squares of Palindromes
100012 = 100020001
110112 = 121242121
111112 = 123454321
112112 = 125686521
| 111 | = 11 |
| 112 | = 121 |
| 113 | = 1331 |
| 114 | = 14641 |
22013 = 10662526601
The only known palindromic cube whose root is not
palindromic !
From Square.htm by Patrick De Geese, Belgium
| 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
8446448
7766
6677
12035788 060
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.
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
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 |
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.
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 at
http://www.oocities.org/~harveyh/primes.htm#3,
5, and 7 Digit Consecutive Primes
and 3 more patterns on my moreprimes page at
http://www.oocities.org/~harveyh/moreprimes.htm#Sophie
Germaine
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
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.
|
LYCHREL NUMBERS: ALL numbers
that do not form a palindrome through the reverse and add process. 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. 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. |
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.
| 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
|
 |
|
|
Two pandiagonal magic squares with the magic sum of 2002.
|
|
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| 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 |
Sophie Germaine related palindrome magic squares |
||||||||||||||||||||
|
2n + 1 => |
|
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By Jaime Ayala. See Carlos Rivera's http://www.primepuzzles.net/puzzles/puzz_124.htm
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)
.Please
send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
September 11, 2009
Copyright © 2002 by Harvey D. Heinz