# 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=11²484=22²676=26²10201=101²12321=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:

 101131151181191 3.13353e+11 7.27758e+11 919929...
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 = 27277x78 = 6006538x539 = 2899821621x1622 = 26292622457x2458 = 6039306 77x78x79 = 474474....

Products Top

 12x42 = 24x2112x63 = 36x2112x84 = 48x2113x62 = 26x31 13x93 = 39x3114x82 = 28x4123x64 = 46x3223x96 = 69x32 24x63 = 36x4224x84 = 48x4226x93 = 39x62 34x86 = 68x4336x84 = 48x6346x96 = 69x64

 2x819 = 9x1823x728 = 8x2734x217 = 7x1244x427 = 7x2444x637 = 7x3644x847 = 7x4845x546 = 6x4556x455 = 5x5467x124 = 4x2177x244 = 4x4277x364 = 4x6378x273 = 3x7289x182 = 2x819 59x25 = 5x295.......... 2x7138 = 83x1724x3149 = 94x134.......... 2198x9 = 9891x23297x8 = 8792x34132x7 = 7231x44264x7 = 7462x44396x7 = 7693x45495x6 = 6594x56594x5 = 5495x67231x4 = 4132x77462x4 = 4264x77693x4 = 4396x78792x3 = 3297x89891x2 = 2198x9.

 1x6264 = 4x6x2611x9168 = 8x6x1912x3168 = 8x6x1323x3464 = 4x6x4334x7866 = 6x6x874.... 3x21525 = 525x1233x42525 = 525x2433x63525 = 525x3633x84525 = 525x4838x22287 = 782x228 8x23575 = 575x328 8x46575 = 575x6488x69575 = 575x968 49x2994 = 499x29459x2995 ​​= 599x29597x6769 = 967x679.....

 144x441 = 252x252156x651 = 273x372168x862 = 294x492276x672 = 384x483. 1224x4221 = 2142x24121236x6321 = 2163x36121248x8421 = 2184x48121584x4851 = 2772x27721596x6951 = 2793x3972 13344x44331 = 23352x2533213356x65331 = 23373x3733213368x86331 = 23394x49332..

Pairs of square numbers Top

 122 = 144 and 212 = 44113² = 169 and 31² = 961.. 102² = 10404 and 201² = 40401103² = 10609 and 301² = 90601112² = 12544 and 211² = 44521113² = 12769 and 3112 = 96721 1012² = 1024144 and 2101² = 44142011112² = 1236544 and 2111² = 44563211212² = 1468944 and 2121² = 44986412012² = 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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