Search for a tool
Lychrel Number

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

Results

Lychrel Number -

Tag(s) : Number Games, Fun/Miscellaneous

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Lychrel Number' tool for free! Thank you!

# Lychrel Number

## Lychrel Number Tester

### What is a Lychrel Number? (Definition)

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).

### How to calculate iterations of a Lychrel number?

To find a Lychrel number:

— Take an initial number ɴ

— Reverse its numbers to find its mirror number

— 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.

### What are the Lychrel numbers?

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

### What is the smallest Lychrel number?

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).

### What is a delayed number?

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.

### What is the the most delayed number?

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

### Why are Lychrel Numbers used?

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.

### What is the code/algorithm for programming Lychrel numbers?

Function testing if a number is a Lychrel number:// Pseudo-codefunction isLychrelNumber(number) { for iteration from 1 to 1000 { number = number + reverse(number) if (number == reverse(number)) return false } return true}// Pythondef 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).

### Is there a number of Lychrel that has been demonstrated?

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.

## Source code

dCode retains ownership of the "Lychrel Number" source code. Except explicit open source licence (indicated Creative Commons / free), the "Lychrel Number" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Lychrel Number" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Lychrel Number" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Lychrel Number" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Lychrel Number on dCode.fr [online website], retrieved on 2024-08-13, https://www.dcode.fr/lychrel-number

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!