Divisor of Perfect Number is Deficient

Theorem

Let $n$ be a perfect number.

Let $n = r s$ where $r$ and $s$ are positive integers such that $r > 1$ and $s > 1$.

Then $r$ and $s$ are both deficient.

Proof

Without loss of generality, consider $r$.

We have by definition of divisor sum function and perfect number that:

$\dfrac {\map {\sigma_1} {r s} } {r s} = 2$

But from Abundancy Index of Product is greater than Abundancy Index of Proper Factors:

$\dfrac {\map {\sigma_1} {r s} } {r s} > \dfrac {\map {\sigma_1} r} r$

That is:

$\dfrac {\map {\sigma_1} r} r < 2$

Mutatis mutandis:

$\dfrac {\map {\sigma_1} r} s < 2$

Hence the result by definition of deficient.

$\blacksquare$

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $12$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $12$