Numeric Palindromes
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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 withthree 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 withfive 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 . . ..

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 . 727 757 787 797 . 919 929 . . .
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 . . . .

Products  top
 12x42 = 24x21 12x63 = 36x21 12x84 = 48x21 13x62 = 26x31 13x93 = 39x31 14x82 = 28x41 23x64 = 46x32 23x96 = 69x32 24x63 = 36x42 24x84 = 48x42 26x93 = 39x62 34x86 = 68x43 36x84 = 48x63 46x96 = 69x64

 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  . . . . . . . . . . 2x7138 = 83x172 4x3149 = 94x134 . . . . . . . . . . . 2198x9 = 9891x2 3297x8 = 8792x3 4132x7 = 7231x4 4264x7 =  7462x4 4396x7 = 7693x4 5495x6 = 6594x5 6594x5 = 5495x6 7231x4 = 4132x7 7462x4 = 4264x7 7693x4 = 4396x7 8792x3 = 3297x8 9891x2 = 2198x9 .

 1x6264 = 4x6x261 1x9168 = 8x6x191 2x3168 = 8x6x132 3x3464 = 4x6x433 4x7866 = 6x6x874 .. . . 3x21525 = 525x123 3x42525 = 525x243 3x63525 = 525x363 3x84525 = 525x483 8x22287 = 782x228  8x23575 = 575x328  8x46575 = 575x648 8x69575 = 575x968 49x2994 = 499x294 59x2995 = 599x295 97x6769 = 967x679 . . . . .

 144x441 = 252x252 156x651 = 273x372 168x862 = 294x492 276x672 = 384x483 . 1224x4221 = 2142x2412 1236x6321 = 2163x3612 1248x8421 = 2184x4812 1584x4851 = 2772x2772 1596x6951 = 2793x3972 13344x44331 = 23352x25332 13356x65331 = 23373x37332 13368x86331 = 23394x49332 .... .

Pairs of Squares top
 12² = 144 and 21² = 441  13² = 169 and 31² = 961 . . 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".

02.02.2020

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 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
Palindromes

Wikipedia
Palindromic numberPalindromic primeEmirp, Lychrel number, Palindrome

Deutsch

Hans-Jürgen Caspar
Palindromzahlen

Jürgen Dankert
Zahlen-Palindrome

Karl Hovekamp

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.

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