The 196 conjecture

Definitions and mathematical statement

For a positive integer $n$ with decimal digits $d_{r-1}\cdots d_0$, define $\operatorname{rev}(n)$ to be the integer represented by the reversed string $d_0\cdots d_{r-1}$; any resulting leading zeros are discarded. Set $R(n)=n+\operatorname{rev}(n)$ and form the sequence $n_0=n$, $n_{j+1}=R(n_j)$. A positive integer is a palindrome when its decimal digit string reads the same forwards and backwards.

Define \[ \mathcal L=\{n\ge1:n_j\text{ is not a palindrome for every }j\ge1\}. \] The source asks two questions: is $\mathcal L$ infinite, and is its least element equal to $196$? The second statement includes both $196\in\mathcal L$ and the assertion that every positive integer smaller than $196$ reaches a palindrome after finitely many positive iterations.

The requirement is avoidance forever. A large number of computed nonpalindromic iterates does not certify membership in $\mathcal L$. The positive-iteration convention is the one in the source: the initial term $n_0$ is not itself tested. Decimal notation is fixed throughout, and neither reversal in another base nor retaining artificial leading zeros changes the operation under consideration. Different starting integers can eventually enter the same orbit, but the infinitude question counts starting integers, not distinct tail-equivalence classes of orbits. Any proof of the claimed minimum must in particular establish nontermination of the reverse-and-add process starting at $196$.

Short English statement

Are infinitely many decimal reverse-and-add sequences forever nonpalindromic, and is $196$ the smallest starting value?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700032 (AMR-046-0032), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 32: The 196 conjecture. Original source indexed by AMR-046-0032. https://arxiv.org/abs/2012.02524.