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`

. – Luis H Gallardo Apr 16 '11 at 0:06`R. Goormaghtigh`

. Google was unable to find something on the former; I have not tried the latter. – Luis H Gallardo Apr 16 '11 at 0:36