# 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).

The palindromes are indicated by black dots.

And so it continues.

Excerpt from the 1000x1000 graph:

**Multiples of 9 **Top

**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

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 palindromes**Top

484=22² 676=26² 10201=101² 12321=111² | 40804=202² 44944=212² 69696=264² 94249=307² | 1002001=1001² 1234321=1111² 4008004=2002² 5221225=2285² | 123454321=11111² . . .. |

**Cubic numbers under the palindromes **Top

343=7³ 1331=11³ 1030301=101³ 1367631=111³

**The prime numbers among the palindromes**Top

All palindromic 3-digit prime numbers:

131 151 181 191 | 353 373 383 . | 757 787 797 . | 929 . . . |

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 numbers**that lead to palindromes top

77x78 = 6006 538x539 = 289982 1621x1622 = 2629262 2457x2458 = 6039306 | . . . . |

**Products **Top

12x63 = 36x21 12x84 = 48x21 13x62 = 26x31 | 14x82 = 28x41 23x64 = 46x32 23x96 = 69x32 | 24x84 = 48x42 26x93 = 39x62 | 36x84 = 48x63 46x96 = 69x64 |

3x728 = 8x273 4x217 = 7x124 4x427 = 7x244 4x637 = 7x364 4x847 = 7x484 5x546 = 6x455 6x455 = 5x546 7x124 = 4x217 7x244 = 4x427 7x364 = 4x637 8x273 = 3x728 9x182 = 2x819 | . . . . . . . . . .
| 4x3149 = 94x134 . | 3297x8 = 8792x3 4132x7 = 7231x4 4264x7 = 7462x4 4396x7 = 7693x4 5495x6 = 6594x5 6594x5 = 5495x6 7231x4 = 4132x7 7462x4 = 4264x7 7693x4 = 4396x7 8792x3 = 3297x8 9891x2 = 2198x9 . |

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

156x651 = 273x372 168x862 = 294x492 276x672 = 384x483 | 1236x6321 = 2163x3612 1248x8421 = 2184x4812 1584x4851 = 2772x2772 1596x6951 = 2793x3972 | 13356x65331 = 23373x37332 13368x86331 = 23394x49332 . . |

**Pairs of square numbers **Top

13² = 169 and 31² = 961 . . | 103² = 10609 and 301² = 90601 112² = 12544 and 211² = 44521 113² = 12769 and 3112 = 96721 | 1112² = 1236544 and 2111² = 4456321 1212² = 1468944 and 2121² = 4498641 2012² = 4048144 and 2102² = 4418404 |

**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.

**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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