Tool for exploring and testing the Lychrel Numbers, these fascinating natural numbers that resist transformation into palindromes by one iteration of mirror number calculation.

Lychrel Number - dCode

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A Lychrel Number is a natural integer that, when subjected to a sequence of mathematical operations (n + reversal of digits of n), never appears to reach a palindrome (a number that reads the same from left to right and right to left).

To find a Lychrel number:

— Take an initial number `ɴ`

— Reverse its numbers to find its mirror number `ᴎ`

— Add the 2 numbers `ɴ+ᴎ`

— Repeat the process with the new number obtained until you obtain a palindrome or until you conclude that the number could be a Lychrel number if no palindrome is found after a significant number of iterations.

__Example:__ `N=360`, its mirror form is `063`, calculation of `360 + 063 = 423`

Start again with `423`: `423 + 324 = 747``747` is a palindrome, so `360` is not a Lychrel number.

All known numbers that form a palindrome do so in less than 300 operations. dCode limits calculations to 500 iterations. If no palindrome is found then the number is probably (but not necessarily) a Lychrel number.

Lychrel numbers are those that never form palindromes. The list of Lychrel numbers is not precisely determined, as it is possible that some numbers will never be proven as such. A conjecture assumes that there are infinitely many of them and therefore that the list is infinite.

Here are the first conjectured Lychrel numbers (below 10000) : 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996, 3493, 3495, 3583, 3585, 3673, 3675, 3763, 3765, 3853, 3855, 3943, 3945, 3995, 4079, 4169, 4259, 4349, 4439, 4492, 4494, 4529, 4582, 4584, 4619, 4672, 4674, 4709, 4762, 4764, 4799, 4852, 4854, 4889, 4942, 4944, 4979, 5078, 5168, 5258, 5348, 5438, 5491, 5493, 5528, 5581, 5583, 5618, 5671, 5673, 5708, 5761, 5763, 5798, 5851, 5853, 5888, 5941, 5943, 5978, 5993, 6077, 6167, 6257, 6347, 6437, 6490, 6492, 6527, 6580, 6582, 6617, 6670, 6672, 6707, 6760, 6762, 6797, 6850, 6852, 6887, 6940, 6942, 6977, 6992, 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, 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, 8795, 8799, 8809, 8850, 8868, 8885, 8889, 8899, 8940, 8958, 8975, 8979, 8989, 8990, 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

See OEIS here

The smallest Lychrel number is 196. However, this is a guess, no one has yet been able to prove that it never forms a palindrome, despite millions of iterations tested.

All numbers before 196 form a palindrome in a few dozen iterations, but it is possible that one day someone will prove that 196 is not a Lychrel number (in which case, probably after billions of iterations).

A delayed number refers to a number that requires a large number of iterations before forming a palindrome. About 90% of numbers form a palindrome in 7 iterations or less.

__Example:__ The number 89 becomes a palindrome after 24 iterations (89 is therefore not a Lychrel number) which makes 89 the most delayed number below 10000.

In 2021, 2 23-digit numbers `13968441660506503386020` and `16909736969870700090800` were discovered after 289 iterations.

Lychrel Numbers are generally not used outside the realm of pure mathematics. They serve primarily as an interesting mathematical case study, but they have no known practical applications.

However, in computer science, Lychrel Numbers are an interesting case for exploring properties of recursion and iterations.

They also arouse interest because of their enigmatic nature (nothing proves that 196 is indeed a Lychrel number) despite their apparent simplicity.

Function testing if a number is a Lychrel number:`// Pseudo-code`

function isLychrelNumber(number) {

for iteration from 1 to 1000 {

number = number + reverse(number)

if (number == reverse(number)) return false

}

return true

}

// Python

def is_lychrel_candidate(n, max_iterations=1000):

for _ in range(max_iterations):

n = n + int(str(n)[::-1])

if (str(n) == str(n)[::-1]):

return False

return True

Assuming that the programming language used already has a `reverse()` function which writes a number backwards (mirrored).

To date, no number has been proven to be definitively a Lychrel number.

The formal proof would require showing that a number can never become a palindrome, regardless of the number of iterations, which is difficult to establish mathematically.

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Cite as source (bibliography):

*Lychrel Number* on dCode.fr [online website], retrieved on 2024-08-13,

- Lychrel Number Tester
- What is a Lychrel Number? (Definition)
- How to calculate iterations of a Lychrel number?
- What are the Lychrel numbers?
- What is the smallest Lychrel number?
- What is a delayed number?
- What is the the most delayed number?
- Why are Lychrel Numbers used?
- What is the code/algorithm for programming Lychrel numbers?
- Is there a number of Lychrel that has been demonstrated?

lychrel,number,palindrome,iteration,delayed

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