C.5 Practice Proof 2

Here’s another more complicated proof for practice. This time, a definition is provided too. Remember: use the self-explanation training after every line you read, either in your head or by writing on paper.

Definition. An abundant number is a positive integer $n$ whose divisors add up to more than $2n$. For example, $12$ is abundant because $1+2+3+4+6+12>24$.

Theorem.The product of two distinct primes is not abundant.

Proof. Let $n=p_{1}p_{2}$, where $p_{1}$ and $p_{2}$ are distinct primes.

Assume that $2\leqslant p_{1}$ and $3\leqslant p_{2}$.

The divisors of $n$ are $1,p_{1},p_{2}$ and $p_{1}p_{2}$.

Note that $\displaystyle\frac{p_{1}+1}{p_{1}-1}$ is a decreasing function of $p_{1}$.

So $\displaystyle\max\left\{\frac{p_{1}+1}{p_{1}-1}\right\}=\frac{2+1}{2-1}=3$.

Hence $\displaystyle\frac{p_{1}+1}{p_{1}-1}\leqslant p_{2}$.

So $p_{1}+1\leqslant p_{1}p_{2}-p_{2}$.

So $p_{1}+1+p_{2}\leqslant p_{1}p_{2}$.

So $1+p_{1}+p_{2}+p_{1}p_{2}\leqslant 2p_{1}p_{2}$. $\square$