Lychrel numbers in base 10

Lychrel conjecture (base 10): conjecturally, there are no Lychrel numbers in base 10. Equivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-lychrel-numbers,
  title        = {Lychrel numbers in base 10},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/lychrel-numbers}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

no_lychrel_numbers_base10. Lychrel conjecture (base 10): conjecturally, there are no Lychrel numbers in base 10.

Equivalently, every positive integer eventually becomes a palindrome under the Lychrel iteration.

isLychrel10_196. The first widely studied open case: whether 196 is a base-10 Lychrel number.

A (base-10) Lychrel number is a positive integer which never becomes a palindrome under the iteration

$a_{0} = n, a_{k + 1} = a_{k} + rev_{10} (a_{k}) .$

One commonly stated conjectural direction is that there are no Lychrel numbers in base 10. The smallest widely studied open case is 196.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.LychrelNumbers (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem no_lychrel_numbers_base10 :
    answer(sorry) ↔ ∀ n : ℕ, 0 < n → ¬ IsLychrel10 n

theorem isLychrel10_196 : answer(sorry) ↔ IsLychrel10 196

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.