Lychrel Number

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.

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, as it has not yet been proven 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.

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

The numbers requiring 0 iterations are the numbers that are already palindromes themselves.

Several numbers complete the algorithm in 1 iteration, some have basic forms that make them predictable:

— All numbers composed only of digits from 0 to 4

— Any number starting (or ending) with 5 and containing only digits from 0 to 4.

— Any number starting (or ending) with 6 and containing only digits from 0 to 3

— Any number starting (or ending) with 7 and containing only digits from 0 to 2

— Any number starting (or ending) with 8 and containing only digits 0 or 1

— Any number starting (or ending) with 9 and containing only the digit 0

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.