Recall from the previous post that:
Equation (A)
$$L(x) + [(x  2)/(x(x  1))] = U(x) + [(x  1)/(x(x + 1))] = 3$$
Furthermore, recall that:
$$L(q) < I(q^k) + I(n^2) <= U(q)$$
where $N = {q^k}{n^2}$ is a (hypothetical) Odd Perfect Number (OPN).
Lastly, recall that:
$$I(q) + I(n^2) = U(q)$$
I have some "pressing" questions at this point:
[[ #1 ]]
What happens to Equation (A) if I change $x$ to $x + 1$? (i.e. What happens if I do a "translation"?)
$$L(x + 1) + [(x  1)/(x(x + 1))] = U(x + 1) + [x/(x + 1)(x + 2))] = 3$$
[[ #2 ]]
What happens to Equation (A) and the last equation if I let
$$x = P$$
and
$$x + 1 = Q$$
(i.e., $P + 1 = Q$)?
(i.e. What happens if I do a "restriction" after doing the "translation" in [[ #1 ]]?)
$$L(P) + [(P  2)/(P(P  1))] = U(P) + [(P  1)/(P(P + 1))] = 3$$
$$L(Q) + [(Q  2)/(Q(Q  1))] = U(Q) + [(Q + 1)/(Q)(Q + 3))] = 3$$
Consequently:
$$L(P) + [(P  2)/(P(P  1))] = U(P) + [(P  1)/(P(P + 1))]$$
equals
$$L(Q) + [(Q  2)/(Q(Q  1))] = U(Q) + [(Q + 1)/(Q)(Q + 3))]$$
[[ #3 ]]
What will happen to the last two equations if you interchange the "roles" of $P$ and $Q$? (In this sense, we are trying to check if $L(x)$ and $U(x)$ are "invertible" functions when restricted to integers $P$ and $Q$ satisfying $Q = P + 1$.)
So we go like:
$$L(Q) + [(Q  2)/(Q(Q  1))] = U(Q) + [(Q  1)/(Q(Q + 1))]$$
$$L(P) + [(P  2)/(P(P  1))] = U(P) + [(P + 1)/(P)(P + 3))]$$
So it does seem to be the case (at least, to myself) that $L(x)$ and $U(x)$ are indeed "invertible".
For the record, the inverses of
$$L(x) = (3x^2  4x + 2)/[x(x  1)]$$
and
$$U(x) = (3x^2 + 2x + 1)/[x(x + 1)]$$
are:
$${L^{1}}(x) = [\sqrt{x^2  8} + (x  4)]/[2(x  3)]$$
$${U^{1}}(x) = [\sqrt{x^2  8}  (x  2)]/[2(x  3)]$$
Note that the inverses satisfy the equation:
$${L^{1}}(x) = {U^{1}}(x) + [(x  1)/(x  3)]$$
or equivalently:
$${L^{1}}(x)  {U^{1}}(x) = 1 + [2/(x  3)]$$
for all $x$ in $[5, \infty)$.
In particular, compare the resulting inequalities for the inverses:
$$1 < {U^{1}}(x) < {L^{1}}(x) < 2$$
with the inequalities for the functions:
$$2\sqrt{2} < L(x) < U(x) < 3$$
IN GENERAL, we have:
$$1 < [U^{1}](x) < [L^{1}](x) < L(x) < U(x) < 3$$
for all $$x \in [5, \infty).$$
LASTLY, note that:
$$L(x + 1) = U(x)$$
for all $x$ not in the set $\{1, 0, 1\}$.
Graphically, we have:
where
= OR ===
both denote the usual equality symbol, and


denotes the usual inequality between the one below and the one above.
(i.e.
$b$


$a$
means that $a < b$).
1 comment:
I am about to post an elementary proof of the OPN conjecture. It will be done in around 13 minutes or so.
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