We show that if
n
≥
3
n \geq 3
is a fixed integer, then there exists an effectively computable constant
c
(
n
)
c (n)
such that if
x
x
,
y
y
, and
m
m
are integers satisfying
x
m
−
1
x
−
1
=
y
n
−
1
y
−
1
,
y
>
x
>
1
,
m
>
n
,
\begin{equation*} \frac {x^m-1}{x-1} = \frac {y^n-1}{y-1}, \; \; y>x>1, \; m > n, \end{equation*}
with
gcd
(
m
−
1
,
n
−
1
)
>
1
\gcd (m-1,n-1)>1
, then
max
{
x
,
y
,
m
}
>
c
(
n
)
\max \{ x, y, m \} > c (n)
. In case
n
∈
{
3
,
4
,
5
}
n \in \{ 3, 4, 5 \}
, we solve the equation completely, subject to this non-coprimality condition. In case
n
=
5
n=5
, our resulting computations require a variety of innovations for solving Ramanujan-Nagell equations of the shape
f
(
x
)
=
y
n
f(x)=y^n
, where
f
(
x
)
f(x)
is a given polynomial with integer coefficients (and degree at least two), and
y
y
is a fixed integer.