Palindrome of numbers

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Number palindromes or palindromic numbers are natural numbers whose number system representation read from the front and back has the same value, e.g. B. 1331 or 742247, but also 21 on base 2 (= 10101). Sometimes the general notation a 1 a 2 a 3 ... | ... a 3 a 2 a 1 is used for numbers with the base .

The term palindrome was adopted from linguistics in number theory , a branch of mathematics .

Palindromes in the decimal system

All numbers in the decimal system with only one digit are palindromic numbers .

There are nine two-digit palindromic numbers:

{11, 22, 33, 44, 55, 66, 77, 88, 99}

There are 90 three-digit palindrome numbers:

{101, 111, 121, 131, 141, 151, 161, 171, 181, 191,
202, 212, 222, 232, 242, 252, 262, 272, 282, 292,
303, 313, 323, 333, 343, 353, 363, 373, 383, 393,
404, 414, 424, 434, 444, 454, 464, 474, 484, 494,
505, 515, 525, 535, 545, 555, 565, 575, 585, 595,
606, 616, 626, 636, 646, 656, 666, 676, 686, 696,
707, 717, 727, 737, 747, 757, 767, 777, 787, 797,
808, 818, 828, 838, 848, 858, 868, 878, 888, 898,
909, 919, 929, 939, 949, 959, 969, 979, 989, 999}

as well as 90 four-digit palindromic numbers:

{1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991,
2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992,
3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993,
4004, 4114, 4224, 4334, 4444, 4554, 4664, 4774, 4884, 4994,
5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995,
6006, 6116, 6226, 6336, 6446, 6556, 6666, 6776, 6886, 6996,
7007, 7117, 7227, 7337, 7447, 7557, 7667, 7777, 7887, 7997,
8008, 8118, 8228, 8338, 8448, 8558, 8668, 8778, 8888, 8998,
9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999}

This means that under 10 4 (i.e. 10,000) there are exactly 9 + 9 + 90 + 90 = 198 number palindromes. In total there are 9 + 9 + 90 + 90 + 900 = 1098 number palindromes that are smaller than 10 5 (i.e. 100,000). The number of palindromes smaller than 10 n follows this series of numbers: 1998 (for n = 6), 10998 (for n = 7 etc.), 19998, 109998, 199998, 1099998, ... ( OEIS , A050250 ).

Furthermore, every whole number that is not divisible by 10 has a positive multiple, which is a decimal palindrome, which had to be proven in a task for the 2009 Federal Mathematics Competition .

The divisibility rules also show that all number palindromes with an even number of digits are divisible by 11.

Generation of number palindromes

Squaring 1's numbers

In the decimal system one gets through

Palindromic numbers, where [1] n is the shorthand notation for the n- fold repetition of 1 and n ranges from 1 to 9.

1 * 1 = 1
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
11111 * 11111 = 123454321
111111 * 111111 = 12345654321
1111111 * 1111111 = 1234567654321
11111111 * 11111111 = 123456787654321
111111111 * 111111111 = 12345678987654321

Inversion and addition

Another possibility is the iterative scheme, in which any positive number (which is not itself a palindrome) is rotated by the following algorithm until it reaches a palindrome :

  1. Reverse the number (e.g. 84 to 48) i.e. H. create the mirror number
  2. Add the upside down number to its starting number (48 + 84 = 132)
  3. Reverse the newly created number again (132 to 231)
  4. Add both numbers again (132 + 231 = 363)

For most numbers, a number palindrome arises after a certain number of calculation steps (up to 10,000 max. 24 steps). However, there are also numbers that oppose this transformation and for which no palindromic formation has yet been found. Such numbers are called Lychrel numbers ; the best known Lychrel number is 196 . The above algorithm is therefore also referred to as the 196 algorithm.

Palindromes when transforming the number system

Number palindromes can also arise when decimal numbers are transformed into another number system.

The following table lists all number palindromes (for numbers from 10 to 10 7 ) that result from the transformation from the decimal system into the respective number system.

Base decimal number Number in another number system
4th 13 31
7th 23 32
46 64
2116 6112
15,226 62,251
8 ( octal ) 1,527,465 5,647,251
9 445 544
313.725 527.313
3,454,446 6,444,543
12 ( duodecimal ) 315.231 132,513
13 43 34
86 68
774 477
14th 834 438
16 ( hexadecimal ) 53 35
371 173
5141 1415
99,481 18,499
19th 21st 12
42 24
63 36
84 48
441 144
882 288
7721 1277
9471 1749
21st 551 155
912 219
22nd 73 37
511 115
25th 83 38
28 31 13
62 26th
93 39
961 169
37 41 14th
82 28
46 51 15th
55 61 16
64 71 17th
73 81 18th
82 91 19th

Sum of number palindromes

In an essay from 2018 it was shown that any positive integer can be written as the sum of three number palindromes, regardless of the number system used with a base of 5 or greater.

See also

literature

Web links

Individual evidence

  1. ^ Federal mathematics competition exercise sheet 2009 1st round. (PDF; 16 kB) Retrieved November 16, 2012 .
  2. ↑ Javier Cilleruelo, Florian Luca, Lewis Baxter: Every positive integer is a sum of three palindromes In: Mathematics of Computation ( arXiv Preprint )