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 palindromesTop
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:
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
77x78 = 6006
538x539 = 289982
1621x1622 = 2629262
2457x2458 = 6039306
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".
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.
(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
Reversible Factors and Multiples
Eric W. Weisstein (MathWorld)
196 Palindrome Quest
Three Years Of Computing (Final Report On The Palindrome Quest, May 25th, 1990)
On General Palindromic Numbers
Patrick De Geest
Palindromic number, Palindromic prime, Emirp, Lychrel number, Palindrome
Palindromic numbers in adic number systems
Collection of word - palindromes
Number palindrome, prime number palindrome, mirp number, Lychrel number, palindrome,
A collection of original and witty gadgets with prime numbers Including palindromes.
This page is also available in German.
URL of my homepage:
© 1999 Jürgen Köller
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