Kourovka Notebook Problem 16.9

Definitions and mathematical statement

Let $F_n$ be the free group with specified free basis $X=\{x_1,\ldots,x_n\}$. For a reduced word $a_1\cdots a_m$, where each $a_i\in X\cup X^{-1}$, its reversal is $a_m\cdots a_1$; reversal does not invert the letters. An element of $F_n$ is an $X$-palindrome if its unique reduced representative equals its reversal. Write $\mathcal P_X$ for the set of these elements, including the identity represented by the empty word. Define the palindromic length by \[ \operatorname{pl}_X(w)=\min\left\{k\in\mathbb Z_{\geq 0}:w=p_1\cdots p_k,\quad p_i\in\mathcal P_X\right\}, \] where the empty product is the identity. This minimum exists because each basis letter and its inverse is a palindrome.

Question. Given an element $w\in F_n$ by a finite word over $X\cup X^{-1}$, is there an algorithm that always terminates and returns $\operatorname{pl}_X(w)$? Equivalently, the input may be required to be reduced.

Unresolved scope qualification. The source gives a finite displayed basis but does not explicitly specify the range of $n$, or whether the rank is fixed in advance. Under the usual positive finite rank interpretation, the fixed-rank formulation asks whether, for every $n\geq 1$, there exists such an algorithm for all words in $F_n$. The uniform formulation asks for a single algorithm taking both $n\geq 1$ and a word in $F_n$ as input. These alternatives are recorded without selecting an intended interpretation.

The supplied registry labels the problem open. Its background also reports an affirmative answer attributed to unverified AI solution claims; those claims are not adopted here as established mathematical results.

Short English statement

Relative to a specified free basis, a palindrome is an element whose reduced word reads identically in either direction. The question asks for an algorithm computing the smallest number of palindromes whose product is a given element. The source leaves uniformity in the rank unspecified and reports unverified affirmative solution claims alongside its open label.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701201$, exact source code catalogue entry, title Kourovka Notebook Problem 16.9. Proposed in 2006 by V. G. Bardakov, V. A. Tolstykh, and V. E. Shpilrain.
• Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), 16th issue (2006), Problem 16.9; also the supplied 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the collection as edited by E. I. Khukhro and V. D. Mazurov. These source attributions have not been independently checked.
• The supplied background labels the following as unverified AI solution claims asserting an affirmative answer: Achyuth Jayadevan, Preprint of 11 September 2026, https://zenodo.org/records/22722079; J. Liao, Preprint of 15 September 2026, https://arxiv.org/abs/2609.17027; M. Kinyon and J. Urban, Preprint of 17 September 2026, https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka-mathematics.pdf.
• Supplied status review dated 2026-10-16: canonical status open, evidence category unrefereed_claim. Its notes attribute a # marker for an unverified AI solution claim to the October 2026 version of the 21st issue, arXiv:1401.0300v48. This is supplied dated curation, not independently verified source or literature assessment. The status-review notes are truncated in the supplied record; the full list of claimed preprints above comes from its background field.