Finding the least critical exponent of sequences is a classical problem of Combinatorics on Words (CoW). The answer is known as Dejean’s conjecture, and the proof was provided step by step by many people. 
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In our talk, we will describe basics from CoW, focusing in particular on repetitions in sequences and the critical exponent.  Then we will describe our recent research concerning the critical exponent of balanced sequences. 
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Recently, Rampersad, Shallit and Vandomme found balanced sequences with the least critical exponent over alphabets of size 3 and 4 and also conjectured that the least critical exponent of balanced sequences over a d-letter alphabet with d>4 is (d-2)/(d-3). Their conjecture was confirmed for d<11. However, we managed to refute their conjecture since we found  balanced sequences over 11 and 12 letters having the critical exponent (d-1)/(d-2). Our research is based on a new algorithm for computation of the critical exponent we have provided and implemented.