Numeric Palindromes
Contents of this Page
What is a Palindrome? 
Counting thePalindromes
Multiples of 9
Strange Equations
Products with the Digit 1
Squares
Cube Numbers
Prime Numbers
Products of  Successive Numbers
Products
Pairs of Squares 
Palindromic Dates
196-Problem
References
Palindromes on the Internet
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To the Main Page    "Mathematische Basteleien"

What is a Palindrome? 
A palindrome is a word, which reads backwards the same as it does forwards. Well known examples are Anna or radar.
You can apply this principle to numbers. For instance 1001 or 69896 are palindromes.

Counting the Palindromes 

All the digits are palindromes (1,2,3,...,9).

There are also 9 palindromes with two digits (11,22,33, ...,99).

You can find to every two-digit number one, and only one number with three digits and with four digits.
For example: For the number 34 there are 343 and 3443.
You can conclude that there are 90 palindromes with three and also 90 palindromes with four digits.

You can find to every three-digit number one, and only one number with five digits and with six digits.
For example: To the number 562 there are 56265 and 562265.
You can conclude that there are 900 palindromes with five and 900 palindromes with six digits.

You have 9+9+90+90+900+900 = 1998 palindromes up to one million. That's 0,1998 %. About every 500th number is a palindrome. 


Position of the Palindromes top
But they are not spread over all numbers regularly. This shows the following picture, which includes the first 1000 numbers.

The numbers are illustrated in a 100x100 square. The numbers are ordered in the same way as you write. One line shows 100 numbers.

And so it goes on.
Part of the 1000x1000-diagram:


Multiples of 9   top

09182736455463728190

Strange Equations   top

(1+2+3+4+5+6+7+8+9+8+7+6+5+4+3+2+1)x12345678987654321 = 999999999²

  2 x (123456789+987654321) +2 = 2222222222

6x7x6 = 252

279972=(2+7+9+9+7+2)x7777


Products with the Digit 1 top

11x11 = 121
111x111 = 12321
1111x1111 = 1234321 
...
111 111 111 x 111 111 111=12345678987654321


11x111 = 1221
111x1111 = 123321
1 111x11111 = 12344321
...
111 111 111 x 1 111 111 111=123456789987654321

I suppose that all products with the digit 1 are palindromes, if one.factor has at the most 9 digits. 
All palindromes have the shape 123.....321.


Squares among the Palindromes   top
121=11²
484=22²
676=26²
10201=101²
12321=111²
14641=121²
 40804=202²
 44944=212²
 69696=264²
 94249=307²
698896=836² 
1002001=1001²
 1234321=1111²
 4008004=2002²
 5221225=2285²
6948496=2636²
 123454321=11111²
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Cube Numbers among the Palindromes top
343=7³                1331=11³       1030301=101³           1367631=111³ 


Prime Numbers among the Palindromes top
All palindromic primes with 3 digits:
101
131
151
181
191
313
353
373
383
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727
757
787
797
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919
929
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There are no primes with 4 digits. They all have the factor 11. (Example:4554=4004+550=4x1001+550=4x91x11+11x50=11x(4x91+50)
There are 93 primes with 5 digits.
There are no primes with 6 digits. They all have the factor 11.
There are 668 primes with 7 digits.


Products of Successive Numbers  top
16x17 =  272
77x78 = 6006
538x539 = 289982
1621x1622 = 2629262
2457x2458 = 6039306
77x78x79 = 474474
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Products  top
2x819 = 9x182
3x728 = 8x273
4x217 = 7x124
4x427 = 7x244
4x637 = 7x364
4x847 = 7x484
5x546 = 6x455
6x455 = 5x546
7x124 = 4x217
7x244 = 4x427
7x364 = 4x637
8x273 = 3x728
9x182 = 2x819
59x25 = 5x295 
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2x7138 = 83x172
4x3149 = 94x134
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2198x9 = 9891x2
3297x8 = 8792x3
4132x7 = 7231x4
4264x7 =  7462x4
4396x7 = 7693x4
5495x6 = 6594x5
6594x5 = 5495x6
7231x4 = 4132x7
7462x4 = 4264x7
7693x4 = 4396x7
8792x3 = 3297x8
9891x2 = 2198x9
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1x6264 = 4x6x261
1x9168 = 8x6x191
2x3168 = 8x6x132
3x3464 = 4x6x433
4x7866 = 6x6x874
..
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3x21525 = 525x123
3x42525 = 525x243
3x63525 = 525x363
3x84525 = 525x483
8x22287 = 782x228
 8x23575 = 575x328
 8x46575 = 575x648
8x69575 = 575x968
49x2994 = 499x294
59x2995 = 599x295
97x6769 = 967x679
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144x441 = 252x252
156x651 = 273x372
168x862 = 294x492
276x672 = 384x483
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1224x4221 = 2142x2412
1236x6321 = 2163x3612
1248x8421 = 2184x4812
1584x4851 = 2772x2772
1596x6951 = 2793x3972
13344x44331 = 23352x25332
13356x65331 = 23373x37332
13368x86331 = 23394x49332
....
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Pairs of Squares   top
12² = 144 and 21² = 441 
13² = 169 and 31² = 961
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102²=10404 and 201²=40401
103²=10609 and 301²=90601
112²=12544 and 211²=44521
113²=12769 and 311²=96721
1012²=1024144 and 2101²=4414201
1112²=1236544 and 2111²=4456321
1212²=1468944 and 2121²=4498641
2012²=4048144 and 2102²=4418404


Palindromic Dates   top
The only palindromic year was 1991 in the last century. 
The only palindromic year is 2002 in this century. If you type in 2002 to a calculator and turn over, 2002 will stay ;-). 


John Will's date of birth: 10 02 2001 (Oct 2, 2001). 

The carnival session 2002/2003 will start on "11.11.2002/11:11". 

Counter on my main page on 2.9.2009, sent by Bernhard Fucyman:

196-Problem    top
Pick a number. Add the number, which you must read from the right to the left (mirror number), to the original number. Maybe the sum is a palindrome. If the sum isn't a palindrome, add the mirror number of the sum to the sum. Maybe you have a palindrome now, otherwise repeat the process. Nearly all numbers have a palindrome in the end. 
Example: 49       49+94=143       143+341= 484 !
There are some numbers, which have no palindromes. The lowest one is 196. But the proof is still missing. 


References   top
(1) Walter Lietzmann, Sonderlinge im Reich der Zahlen, Bonn, 1947
(2) Walter Sperling, Auf du und du mit Zahlen, Rüschlikon-Zürich, 1955
(3) Erwein Flachsel, Hundertfünfzig Mathe-Rätsel, Stuttgart 1982, Seite 138 f. 
(4) Martin Gardner, Mathematischer Zirkus, Berlin 1988, Seite 259 ff.


Palindromes on the Internet     top

English

Chip Burkitt
Reversible Factors and Multiples

Jason Allen Doucette
196 Palindrome Quest

John Walker
Three Years Of Computing (Final Report On The Palindrome Quest, May 25th, 1990) 

MathPages
On General Palindromic Numbers

Patrick De Geest
Palindromes

Peter Collins
Fascinating Palindromes

Wade VanLandingham
196-Problem

Wikipedia
Palindromic numberPalindromic primeEmirp, Lychrel number, Palindrome



Deutsch

Jürgen Dankert 
Zahlen-Palindrome

Karl Hovekamp
Palindromzahlen in adischen Zahlensystemen

Ulf Hinze
Sammlung von Wort - Palindromen

Wikipedia
Zahlenpalindrom, Primzahlpalindrom, Mirpzahl, Lychrel-Zahl, Palindrom

Willi Jeschke
Eine Sammlung origineller und geistreicher Spielereien mit Primzahlen Darunter sind auch Palindrome.

Winfrid Krone 
Strukturen in einigen künstlichen Zahlenfolgen


Feedback: Email address on my main page

This page is also available in German.

URL of my Homepage:
http://www.mathematische-basteleien.de/

©  1999 Jürgen Köller

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