Prove that $p^j q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.

Define $\sigma(m) = \sum$ d : d|n. Prove that $p^j$$q^i$ cannot be a perfect number for $p, q$ odd, distinct primes.

Attempt at Solution: I have shown that $p^k$ can never be a perfect number, and im trying to use the multiplicative property of $\sigma$ to generalize to $p^j$$q^i$

Solution 1:

We have $$\sigma(p^jq^i)=\sigma(p^j)\sigma(q^i)=\frac{p^{j+1}-1}{p-1}\cdot \frac{q^{i+1}-1}{q-1}\lt \frac{p^{j+1}q^{i+1}}{(p-1)(q-1)}=p^jq^i \frac{p}{p-1}\cdot \frac{q}{q-1}.$$ Without loss of generality we can assume that $p\lt q$. Note that $\frac{p}{p-1}\le \frac{3}{2}$ and $\frac{q}{q-1}\le \frac{5}{4}$. It follows that $\frac{p}{p-1}\cdot \frac{q}{q-1}\le \frac{15}{8}\lt 2$. We conclude that $p^jq^i$ is deficient, and in particular cannot be perfect.