What is 1 111 111 111 squared


What is a palindrome?
A palindrome is usually a word that stays the same even when read from right to left. Well-known words are Otto, Anna, Reliefpfeiler or Pensioner.
This property can also be transferred to numbers. 1001 or 69896 are palindromes.

Number of palindromes Top

All 9 single digit numbers 1 to 9 are palindromes.

There are also 9 two-digit palindromes (11,22,...99).

For each two-digit number you can create a three-digit and a four-digit palindrome.
(E.g. for the number 34 there are 343 and 3443)
So there are 90 three-digit palindromes and also 90 four-digit palindromes.

For each three-digit number you can create a one-to-one five and six-digit palindrome.
(E.g. for the number 562 there are 56265 and 562265.)
So there are 900 five-digit palindromes and also 900 six-digit palindromes.

Under 1 million there are 9 + 9 + 90 + 90 + 900 + 900 = 1998 palindromes.
That is 0.1998% of all numbers. About every 500th number is a palindrome.


Distribution of the palindromes Top
The palindromes are not evenly distributed. This is shown in the following diagram, which records the first 10,000 numbers (including 198 palindromes).

In the 100x100 picture, the numbers from 1 to 10,000 are represented by a square of 4 pixels. You go through the numbers from top left to bottom right as you write. After every 100 numbers, the new line continues.
The palindromes are indicated by black dots.

And so it continues.
Excerpt from the 1000x1000 graph:


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 ones Top

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


11x111 = 1221
111x1111 = 123321
1 111x11111 = 12344321
...
111 111 111x1 111 111 111 = 123456789987654321

I suspect that all products of numbers with 1 are palindromes as long as a factor has 9 or fewer digits. All palindromes have the representation 123 .......... 321.


The square numbers under the palindromesTop

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
.
.
..

Cubic numbers under the palindromes Top
343=7³                1331=11³       1030301=101³           1367631=111³ 


The prime numbers among the palindromesTop
All palindromic 3-digit prime numbers:

101
131
151
181
191
313
353
373
383
.
727
757
787
797
.
919
929
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.
.
There are no 4-digit palindromic prime numbers. You have the divisor 11. (Example: 4554 = 4004 + 550 = 4x1001 + 550 = 4x91x11 + 11x50 = 11x (4x91 + 50)
There are 93 5-digit palindromic prime numbers.
There are no 6-digit palindromic prime numbers. You have the divisor 11.
There are 668 7-digit palindromic primes.

Products of neighboring numbersthat lead to palindromes top

16x17 = 272
77x78 = 6006
538x539 = 289982
1621x1622 = 2629262
2457x2458 = 6039306
77x78x79 = 474474
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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
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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
.

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 square numbers 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
Benedikt Plasa, thank you for the hint.

Palindromic data Top
In the last century the year was 1991, in this century the year 2002 is always the only palindromic year. If you put the number 2002 in the calculator, it stays there even if you turn the calculator upside down.
The next palindromic year will be 2112.


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

In the company "Valenzia - Karl-H.Vogt in D29556 Suderburg" there is a palindromic sense of humor: Your wild-lingonberry selection was good until 11.11.2002 / 11:11.

The 2002/2003 carnival season begins on "11.11.2002 / 11:11".

02.02.2020

Counter on my main page on September 2, 2009, sent by Bernhard Fucyman:

196 problem Top
Pick any number. Add the number read from right to left (mirror number) to the original number. Maybe the sum is a palindrome. If not, add the mirror number of the sum to the sum. Perhaps a palindrome has now arisen. If not, repeat the process.
Almost all numbers have a palindrome at the end.
Example: 49 49 + 94 = 143 143 + 341 = 484!
There are quite a few numbers that don't seem to have a palindrome. The smallest number is 196. A mathematical proof is still missing.


credentials Top
(1) Walter Lietzmann, Oddities in the Realm of Numbers, Bonn, 1947
(2) Walter Sperling, On you and you with numbers, Rüschlikon-Zurich, 1955
(3) Erwein Flachsel, Hundred and Fifty Math Riddles, Stuttgart 1982, page 138 f.
(4) Martin Gardner, Mathematischer Zirkus, Berlin 1988, page 259 ff.


Palindromes on the Internet Top

English

Chip Burkitt
Reversible Factors and Multiples

Eric W. Weisstein (MathWorld)
Palindromic Number

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 number, Palindromic prime, Emirp, Lychrel number, Palindrome



German

Hans-Jürgen Caspar
Palindromic numbers
 
Jürgen Dankert
Number palindromes

Karl Hovekamp
Palindromic numbers in adic number systems

Ulf Hinze
Collection of word - palindromes

Wikipedia
Number palindrome, prime number palindrome, mirp number, Lychrel number, palindrome,

Willi Jeschke
A collection of original and witty gadgets with prime numbers Including palindromes.


I would like to thank Benjamin Böck for the contributions: Diagrams, number of 7-digit palindromic prime numbers

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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