Processing math: 100%

Search This Blog

5.5.18

Can the following argument be pushed to a full proof that (p + 2)/p is an outlaw if p is an odd prime?

(The following post is extracted verbatim from this MSE question.)

This is related to this earlier MSE question.

Let \sigma(x) be the sum of the divisors of x, and denote the abundancy index of x by I(x):=\sigma(x)/x.

If the equation I(a) = b/c has no solution a \in \mathbb{N}, then the rational number b/c is said to be an abundancy outlaw.

In the earlier question, it is shown that:

(a) If I(n) = (p+2)/p, then n is an odd square.

(b) If I(n) = (p+2)/p, then p \mid n.

So now suppose to the contrary that I(n) = (p+2)/p.  From (a) and (b), since p is a prime number, then p^2 \mid n.  It follows that
\frac{p^2 + p + 1}{p^2} = I(p^2) \leq I(n) = \frac{p+2}{p},
whence there is still no contradiction.

From the divisibility constraint p^2 \mid n, we have that p^2 \leq n.  We claim that p^2 \neq n.  Suppose to the contrary that p^2 = n.

Then we have
\frac{\sigma(n)}{n} = \frac{p+2}{p} = \frac{p(p+2)}{p^2}
so that
p^2 + p + 1 = \sigma(p^2) = \sigma(n) = p^2 + 2p,
which contradicts the fact that p is an odd prime.

This implies that p < \sqrt{n}, which further means that
\gcd(n, \sigma(n)) = \frac{n}{p} > \sqrt{n} > {10}^8
where the lower bound n > {10}^{16} is due to Richard F. Ryan, "Results concerning uniqueness for \sigma(x)/x = \sigma(p^n q^m)/(p^n q^m) and related topics, International Math. J., 2002 , V2#5pp497-514".

Here is my question:

Can the preceding argument be pushed to a full proof that (p + 2)/p is an outlaw if p is an odd prime?