Is there a generalisation of Catalan's Conjecture hiding in the solution to $(2^m-1) = (2^n-1)k^2$?
I ask having noticed the similarity between the answer to this; Solve the equation $(2^m-1) = (2^n-1)k^2$ and the conjecture. Mercury's nontrivial solution is:
$3^2(2^3-1)=2^6-1$
$3^2(2^3-1)=(1+2^3)(2^3-1)$
Dividing throughout by $2^3-1$ we have the sole example of the Catalan statement:
$3^2=2^3-1$
Which suggests to me Mercury's answer may be a generalisation or special case of the (already proven) Catalan's conjecture.
The original question might therefore be generalised as follows:
Let $(2^{pn}-1) = (2^n-1)k^r$
Do $p,n,k,r$ in general obey Catalan's conjecture:
$k^r-p^n=1$
This might be thought of intuitively as; we know $3^2$ is the only perfect square ratio between two Mersenne numbers $M_q$ (where $q$ is not necessarily prime), but is it further the only perfect power?
And generalising further we might conjecture the same identity holds for:
$(p^{pn}-1) = (p^n-1)k^r$