LocalGlobal < CoWiki

CoWiki Web>AdditionalOpenProblems>LocalGlobal (2018-02-22, JeffreyShallit) (raw view)

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 200€ (<strike>100 €</strike> upped during WORDS 2011 in Prague) is promised. [[http://www.karlin.mff.cuni.cz/~holub/soubory/prizeproblem.pdf][Details here.]]

Recently, the problem has been solved by Aleksi Saarela.

-- Main.StepanHolub - 01 Aug 2011

Topic revision: r4 - 2018-02-22 - JeffreyShallit

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