LocalGlobal < CoWiki

Is there a positive integer n &ge; 2 and words u<sub>1</sub>, u<sub>2</sub>, ...,  u<sub>n</sub> such that
both equalities

(u<sub>1</sub>u<sub>2</sub> &sdot;&sdot;&sdot; u<sub>n</sub>)<sup>2</sup> = u<sub>1</sub><sup>2</sup>u<sub>2</sub><sup>2</sup> &sdot;&sdot;&sdot; u<sub>n</sub><sup>2</sup>,

(u<sub>1</sub>u<sub>2</sub> &sdot;&sdot;&sdot; u<sub>n</sub>)<sup>3</sup> = u<sub>1</sub><sup>3</sup>u<sub>2</sub><sup>3</sup>&sdot;&sdot;&sdot; u<sub>n</sub><sup>3</sup>

hold simultaneously and the words u<sub>i</sub>, i = 1, ..., n, do not pairwise commute (that
is, u<sub>i</sub>u<sub>j</sub> &ne; u<sub>j</sub>u<sub>i</sub> for at least one pair of indices i,j&isin;{1,2,...,n })?

For the solution an award of 100 &euro; is promised. [[http://www.karlin.mff.cuni.cz/~holub/soubory/prizeproblem.pdf][Details here.]]


-- Main.StepanHolub - 01 Aug 2011

This topic: CoWiki > WebHome > AdditionalOpenProblems > LocalGlobal
Topic revision: r2 - 2011-08-05 - JeffreyShallit