Let
W
=
W
(
x
1
,
…
,
x
j
)
W = W({x_1}, \ldots ,{x_j})
be any word in the
j
j
free generators
x
1
,
…
,
x
j
{x_1}, \ldots ,{x_j}
, and suppose that
W
W
cannot be expressed in the form
W
=
V
k
W = {V^k}
for
V
V
a word and for
k
k
an integer with
|
k
|
≠
1
\left | k \right | \ne 1
. We ask whether the equation
f
=
W
f = W
has a solution
(
x
1
,
…
,
x
j
)
=
(
a
1
,
…
,
a
j
)
∈
G
j
({x_1}, \ldots ,{x_j}) = (a_{1}, \ldots , a_{j}) \in G^{j}
whenever
G
G
is an infinite symmetric group and
f
f
is an element in
G
G
. We establish an affirmative answer in the case that
W
(
x
,
y
)
=
x
m
y
n
W(x,y) = {x^m}{y^n}
for
m
m
and
n
n
nonzero integers.