16.83 (2006)
Partially SolvedLet $E_n$ be a free locally nilpotent $n$-Engel group on countably many generators, and let $\pi(E_n)$ be the set of prime divisors of the orders of elements of the periodic part of $E_n$. It is known that $2, 3, 5 \in \pi(E_4)$.
$\qquad$ a) Does there exist $n$ for which $7 \in \pi(E_n)$?
$\qquad$ b) Is it true that $\pi(E_n) = \pi(E_{n+1})$ for all sufficiently large $n$?
Progress
a) Yes, n = 6, since the 2-generator free nilpotent 6-Engel group has an element of order 7 (W. Nickel, J. Austral. Math. Soc., Ser. A, 67, no. 2 (1999), 214–222; http://www.mathematik.tu-darmstadt.de/~nickel/Engel/Engel.html) (A. Abdollahi, Letter of 19 August 2011).
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