Lychrel number

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The Lychrel numbers are certain natural numbers that oppose the formation of the palindrome by a certain algorithm , the 196 algorithm .

The name Lychrel comes from Wade VanLandingham and has no particular meaning, except that prior to naming these numbers, Google did not return a search result for Lychrel and it was not listed in any dictionary, and it is an approximate anagram to the name of VanLandingham's friend (" Cheryl ").

Properties of Lychrel Numbers

Every natural number that does not lead to a number palindrome through a finite number of inversions and additions is called a Lychrel number. Inversion is understood here as the formation of the mirror-inverted number . If the addition leads to a palindrome of numbers, the algorithm is ended. If not, this process is carried out by renewed inversion and addition until the result is a palindrome.

Examples

  • Take the number 5273. The reverse number is 3725 (inversion). The number palindrome 8998 is obtained by adding.
  • For other numbers, this algorithm takes longer:
4753 + 3574 = 8327
8327 + 7238 = 15565
15565 + 56551 = 72116
72116 + 61127 = 133243
133243 + 342331 = 475574 (a palindrome)
  • The Lychrel numbers resist palindromic formation. The smallest Lychrel number is probably the number 196. So far, there is no mathematical proof that, based on 196, the inversion will definitely never result in a palindrome. Even the very large number of calculated iterations (almost 725 million ) does not allow any statement about the validity of this assumption. See below .

Records

According to the number of iterations, with the lowest possible starting number

(Starting number less than 100,000, excluding candidates for Lychrel numbers)

number ITER
ation 		palindrome
Put number
1 1 1 2
5 2 2 11
59 3 4th 1,111
69 4th 4th 4,884
79 6th 5 44,044
89 24 13 8.813.200.023.188
10,548 30th 17th 17,858,768,886,785,871
10,677 53 28 4,668,731,596,684,224,866,951,378,664
10,833 54 28 4,668,731,596,684,224,866,951,378,664
10,911 55 28 4,668,731,596,684,224,866,951,378,664

The record is currently at 261 iteration steps, this requires the number 1,186,060,307,891,929,990 (19 digits) to arrive at a 119-digit palindrome.

calculation

So far, the algorithm has been applied to all numbers up to 18-digit, and by January 2010 it was also applied to 55 percent of all 19-digit numbers (excluding candidates for Lychrel numbers).

Lychrel numbers

The Lychrel numbers defied this algorithm, which means that - even after an infinite number of iterations - no palindrome arises.

At the moment there is no mathematical method to determine with certainty whether a number is a Lychrel number, so that to this day it is not even certain whether they even exist.

Smallest candidate found for the Lychrel numbers

The smallest number that the 196 algorithm has not yet been able to convert into a palindrome is 196 (hence the name 196 algorithm). Since this is the first Lychrel candidate, this number has been the best studied to date. By May 1, 2006, Wade VanLandingham electronically calculated 724,756,966 iterations based on the 196. The last result number had 300,000,000 digits and was still not a palindrome. The calculation began in August 2001 and lasted almost five years, although it must be mentioned here that you could fall back on a calculation that had already been carried out up to a 14,000,000-digit result (33,824,775 iterations), the first results of which were already carried out In the early 1990s .

Candidates for which Lychrel numbers are less than 10,000
  • between 1 and 999:
    196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986
  • between 1000 and 1999:
    1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997
  • between 2000 and 2999:
    2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996
  • between 3000 and 3999:
    3493, 3495, 3583, 3585, 3673, 3675, 3763, 3765, 3853, 3855, 3943, 3945, 3995
  • between 4000 and 4999:
    4079, 4169, 4259, 4349, 4439, 4492, 4494, 4529, 4582, 4584, 4619, 4672, 4674, 4709, 4762, 4764, 4799, 4852, 4854, 4889, 4942, 4944, 4979 , 4994
  • between 5000 and 5999:
    5078, 5168, 5258, 5348, 5438, 5491, 5493, 5528, 5581, 5583, 5618, 5671, 5673, 5708, 5761, 5763, 5798, 5851, 5853, 5888, 5941, 5943, 5978 , 5993
  • between 6000 and 6999:
    6077, 6167, 6257, 6347, 6437, 6490, 6492, 6527, 6580, 6582, 6617, 6670, 6672, 6707, 6760, 6762, 6797, 6850, 6852, 6887, 6940, 6942, 6977 , 6992
  • between 7000 and 7999:
    7059, 7076, 7149, 7166, 7239, 7256, 7329, 7346, 7419, 7436, 7491, 7509, 7526, 7581, 7599, 7616, 7671, 7689, 7706, 7761, 7779, 7796, 7851 , 7869, 7886, 7941, 7959, 7976, 7991
  • between 8000 and 8999:
    8058, 8075, 8079, 8089, 8148, 8165, 8169, 8179, 8238, 8255, 8259, 8269, 8328, 8345, 8349, 8359, 8418, 8435, 8439, 8449, 8490, 8508, 8525 , 8529, 8539, 8580, 8598, 8615, 8619, 8629, 8670, 8688, 8705, 8709, 8719, 8760, 8778, 8795, 8799, 8809, 8850, 8868, 8885, 8889, 8899, 8940, 8958, 8975 , 8979, 8989, 8990
  • between 9000 and 9999:
    9057, 9074, 9078, 9088, 9147, 9164, 9168, 9178, 9237, 9254, 9258, 9268, 9327, 9344, 9348, 9358, 9417, 9434, 9438, 9448, 9507, 9524, 9528 , 9538, 9597, 9614, 9618, 9628, 9687, 9704, 9708, 9718, 9777, 9794, 9798, 9808, 9867, 9884, 9888, 9898, 9957, 9974, 9978, 9988

Distribution of candidates for Lychrel numbers
Number of
candidates		 Number space
013 0000 to 0999
013 1000 to 1999
013 2000 to 2999
013 3000 to 3999
024 4000 to 4999
024 5000 to 5999
024 6000 to 6999
029 7000 to 7999
051 8000 to 8999
045 9000 to 9999
249 0000 to 9999

The first iterations for small candidates

The rows list candidates for Lychrel numbers, with numbers on top of each other being inverses of each other. Behind it is the total sum of all pairs.

  • 196, 295, 394
    691, 592, 493 • 790 → 887
  • 689, 788
    986, 887 → 1675
  • 1495, 1585, 1675 , 1765, 1855, 1945, 2494, 2584, 2674, 2764, 2854, 2944, 3493, 3583, 3673
    5941, 5851, 5761, 5671, 5581, 5491, 4942, 4852, 4762, 4672, 4582 , 4492, 3943, 3853, 3763 • 6490, 6580, 6670, 6760, 6850, 6940 → 7436
  • 4079, 4169, 4259, 4349, 4439, 4529, 4619, 4709, 5078, 5168, 5258, 5348, 5438, 5528, 5618, 5708, 6077, 6167, 6257, 6347, 6437, 6527, 6617, 6707
    9704, 9614 , 9524, 9434, 9344, 9254, 9164, 9074, 8705, 8615, 8525, 8435, 8345, 8255, 8165, 8075, 7706, 7616, 7526, 7436 , 7346, 7256, 7166, 7076 → 13783


  • 4799, 4889, 4979, 5798, 5888, 5978, 6797, 6887, 6977
    9974, 9884, 9794, 8975, 8885, 8795, 7976, 7886, 7796 → 14773


  • 7059, 7149, 7239, 7329, 7419, 7509, 8058, 8148, 8238,
    9507, 9417, 9327, 9237, 9147, 9057, 8508, 8418, 8328 → 16566


  • 7599, 7689, 7779, 7869, 7959, 8598, 8688
    9957, 9867, 9777, 9687, 9597, 8958, 8868 → 17556


  • 879
    978 → 1857
  • 1497, 1587, 1677, 1767, 1857 , 1947, 2496, 2586, 2676, 2766, 2856, 2946, 3495, 3585, 3675, 3765, 3855, 3945, 4494, 4584, 4674
    7941, 7851, 7761, 7671, 7581 , 7491, 6942, 6852, 6762, 6672, 6582, 6492, 5943, 5853, 5763, 5673, 5583, 5493, 4944, 4854, 4764 • 8490, 8580, 8670, 8760, 8850, 8940 → 9438
  • 8079, 8169, 8259, 8349, 8439, 8529, 8619, 8709
    9708, 9618, 9528, 9438 , 9348, 9258, 9168, 9078 → 17787


  • 8089, 8179, 8269, 8359, 8449, 8539, 8629, 8719, 8809
    9808, 9718, 9628, 9538, 9448, 9358, 9268, 9178, 9088 → 17897


  • 8799, 8889, 8979
    9978, 9888, 9798 → 18777


  • 1997, 2996, 3995
    7991, 6992, 5993 • 8990 → 9988
  • 8899, 8989
    9988 , 9898 → 18887

Web links

Individual evidence

  1. ↑ a b www.jasondoucette.com "Weltrekorde", section Most Delayed Palindromic Number Records
  2. ↑ Website about the Lychrel numbers ( memento of the original dated November 4, 2006 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.p196.org
  3. ↑ Follow A023108 in OEIS (English)
  4. ↑ a b http://www.p196.org/files.html ( Memento of the original from May 1, 2009 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. @1@ 2Template: Webachiv / IABot / www.p196.org
  5. ↑ rosettacode.org