The following equation
$$ \sigma(P^n)=\sigma(Q^m) $$
has only the trivial solution
$$ P=Q, m=n, $$
for $P,Q$ irreducible polynomials of degree $>1$ in the ring $$ GF(2)[t] $$
where for any $A \in GF(2)[t]$
$$ \sigma(A) = \sum_{d \mid A} d $$
Question: What happens over the integers. More precisely:
What are the solutions (besides $p=q,m=n$) (or what can be said about the solutions, besides $m \equiv n \pmod{2}$ ) of the equation
$$ \sigma(p^n) = \sigma(q^m) $$
for odd prime numbers $p,q$ and positive integers $m,n$
where
$$
\sigma(n) = \sum_{0<d, d \mid n} d.
$$
Diophantine Equations
. $\endgroup$R. Goormaghtigh
. Google was unable to find something on the former; I have not tried the latter. $\endgroup$