Lychrel number

The Lychrel numbers are certain natural numbers that defy the Palindrombildung by a certain algorithm, the algorithm 196.

The name Lychrel comes from Wade VanLandingham and has no special meaning, except that prior to the designation of these figures, Google gave no result for Lychrel and it was not listed in any dictionary, and there is an approximate anagram for the name of the girlfriend of VanLandingham ( " Cheryl ").

Properties of Lychrel numbers

Any natural number that does not perform a finite number of additions and inversions to any of the figures palindrome is called Lychrel number. When inversion is meant here the formation of mirror-image number. It performs the addition to a Zahlenpalindrom, the algorithm is terminated. If not, this process is carried out until the result is a palindrome by repeated inversion and addition.

Examples

  • Take the number 5273rd The mirrored to number 3725 is ( inversion). By addition we obtain the Zahlenpalindrom 8998th
  • For other numbers, this algorithm takes longer:
  • The Lychrel numbers oppose the Palindrombildung. The smallest Lychrel number is probably the number 196 A mathematical proof that, starting from 196, the inversion definitely never a palindrome is found but does not exist, yet. The very large number of calculated iterations ( almost 725 million ) does not allow any statement about the validity of this assumption. See below.

Records

( Initial number less than 100,000, candidates for Lychrel numbers excluded)

The record currently stands at 261 iterations, this requires the number 1.186.060.307.891.929.990 (19 points ), to get to a 119 - digit palindrome.

Calculation

So far, the algorithm at all to 18 - digit numbers was carried out to date (5 April 2009) he was also applied at 43.083 percent of all 19-digit numbers ( candidates for Lychrel numbers excluded).

Lychrel numbers

The Lychrel numbers opposed this algorithm, which means that - even after infinitely many iterations - not a palindrome is formed.

Currently, there is no mathematical method to determine certain whether a number is a Lychrel number, so that is still not even sure if they exist at all.

The smallest number that has not yet been converted by the algorithm 196 in a palindrome, is 196 ( hence the name 196 algorithm). Since this is the first candidate Lychrel, this number is previously studied the best. Until May 1, 2006 computed Wade VanLandingham electronic 724 756 966 iterations, starting from the 196th The final result number was 300,000,000 points and was still not a palindrome. The calculation began in August 2001 and lasted almost five years, and here it must be mentioned that one could rely on an already already calculation performed up to a 14,000,000 -digit earnings ( 33,824,775 iterations performed ) at the time, the first results already early 1990s were calculated.

  • Between 1 and 999: 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986
  • 1000-1999: 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997
  • 2000-2999: 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996
  • 3000-3999: 3493, 3495, 3583, 3585, 3673, 3675, 3763, 3765, 3853, 3855, 3943, 3945, 3995
  • 4000-4999: 4079, 4169, 4259, 4349, 4439, 4492, 4494, 4529, 4582, 4584, 4619, 4672, 4674, 4709, 4762, 4764, 4799, 4852, 4854, 4889, 4942, 4944, 4979, 4994
  • 5000-5999: 5078, 5168, 5258, 5348, 5438, 5491, 5493, 5528, 5581, 5583, 5618, 5671, 5673, 5708, 5761, 5763, 5798, 5851, 5853, 5888, 5941, 5943, 5978, 5993
  • 6000-6999: 6077, 6167, 6257, 6347, 6437, 6490, 6492, 6527, 6580, 6582, 6617, 6670, 6672, 6707, 6760, 6762, 6797, 6850, 6852, 6887, 6940, 6942, 6977, 6992
  • 7000-7999: 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
  • 8000-8999: 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
  • 9000-9999: 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, 9999

The lines show respectively on candidates for Lychrel - numbers, one above the other standing figures each are inverses of each other. Behind this is the common sum of all couples.

  • 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
Uniform Resource Locator
536715
de