12.64 (1992)

Is it true that for a given number $k \geqslant 2$ and for any (prime) number $n$, there exists a number $N = N(k, n)$ such that every finite group with generators $A = \{a_1, \dots, a_k\}$ has exponent $\leqslant n$ if $(x_1 \dots x_N)^n = 1$ for any $x_1, \dots, x_N \in A \cup \{1\}$?

For given $k$ and $n$, a negative answer implies, for example, the infiniteness of the free Burnside group $B(k, n)$, and a positive answer, in the case of sufficiently large $n$, gives, for example, an opportunity to find a hyperbolic group which is not residually finite (and in this group a hyperbolic subgroup of finite index which has no proper subgroups of finite index).