Summary of Results on Odd Perfect Numbers - As of April 3 2013 - Arnie B. Dris

** Updated April 15 2013 - 9:00 PM Manila time **

Let $N = {q^k}{n^2}$ be an Odd Perfect Number (hereinafter abbreviated as OPN) given in Eulerian form (i.e., $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$).  Note that $\gcd(q, n) = 1$ forces $q \neq n$ and $q^k \neq n$.

The following conjecture was made in 2008:

Conjecture 1 [Dris - 2008]  If $N = {q^k}{n^2}$ is an OPN given in Eulerian form, then $q^k < n$.

This conjecture was made on the basis of the following observation:

Observation 1 [Dris - 2008]  If $N = {q^k}{n^2}$ is an OPN given in Eulerian form, then $I(q^k) < I(n)$, where $I(x) = \displaystyle\frac{\sigma(x)}{x}$ is the abundancy index of the (positive) integer $x$.

When the author made the said conjecture in the year 2008, he was under the naive impression that the divisibility constraint $\gcd(q, n) = 1$ induced an "ordering property" between $q^k$ and $n = \sqrt{\displaystyle\frac{N}{q^k}}$ (in the sense that, similarly, the inequality $q^k < n^2$ [Dris - 2008] appears to follow from the related inequality $I(q^k) < I(n^2)$.)

Indeed, we have the following inequalities:

Lemma 1 [Dris - 2012]  If $N = {q^k}{n^2}$ is an OPN given in Eulerian form, then $I(q^k) < \sqrt[3]{2} < I(n)$ and $I(q^k) < \sqrt{2} < I(n^2)$.

From Lemma 1, we have the following (slight) improvement to the second inequality:

Corollary 1 [Dris - 2013 (currently a work in progress)]  If $N = {q^k}{n^2}$ is an OPN given in Eulerian form, then $\left(I(q^k)\right)^2 < \frac{25}{16} < \frac{8}{5} < I(n^2)$.

The proof of Corollary 1 follows from observing that $\frac{2}{I(n^2)} = I(q^k) < \frac{5}{4}$, if $N = {q^k}{n^2}$ is an OPN given in Eulerian form.

At this point, we dispose of the following lemma:

Lemma 2 [Dris - 2013 (currently a work in progress)]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  Then at least one of the following sets of inequalities is true:

$$q^k < \sigma(q^k) < n < \sigma(n)$$
$$n < q^k < \sigma(q^k) < \sigma(n)$$
$$n < \sigma(n) < q^k < \sigma(q^k)$$
$$q^k < n < \sigma(q^k) < \sigma(n)$$
$$n < q^k < \sigma(n) < \sigma(q^k)$$


Lemma 2 is proved by listing all possible permutations of the multi-set $\{q^k, \sigma(q^k), n, \sigma(n)\}$, subject to the (trivial) abundancy index constraints $1 < I(q^k) < I(n)$.

There are two underlying cases in Lemma 2:

Case 1:
$$\frac{\sigma(q^k)}{n} < \frac{\sigma(n)}{q^k}$$

Case 2:
$$\frac{\sigma(n)}{q^k} < \frac{\sigma(q^k)}{n}$$

Since $I(q^k) < I(n)$ implies $\displaystyle\frac{\sigma(q^k)}{\sigma(n)} < \displaystyle\frac{q^k}{n}$, then we have the following implications:

A.  Case 1 implies that $\sigma(q^k) < \sigma(n)$.

B.  Case 2 implies that $n < q^k$.

At this point, recall that Sorli's conjecture (made in the year 2003) for OPNs predicts that $k = \nu_{q}(N) = 1$, if $N = {q^k}{n^2}$ is an OPN given in Eulerian form.

Additionally, note from Lemma 2 that if either one (but not both) of the following scenarios hold:

$$q^k < n < \sigma(q^k) < \sigma(n)$$
$$n < q^k < \sigma(n) < \sigma(q^k)$$

then Sorli's conjecture cannot follow (i.e., $k \neq 1$), because these scenarios would then violate the fact that $q$ and $\sigma(q) = q + 1$ are consecutive (positive) integers.

If we consider the contrapositive of the last statement, we get the following proposition:

$$k = 1 \Longrightarrow \{(n < q^k) \vee (\sigma(q^k) < n) \vee (\sigma(n) < \sigma(q^k))\} \\ \vee \{(q^k < n) \vee (\sigma(n) < q^k) \vee (\sigma(q^k) < \sigma(n)\},$$

since it is always the case that $q^k < \sigma(q^k)$ and $n < \sigma(n)$ (i.e., $I(x) > 1 \forall x > 1$).  The conclusion in this implication can be simplified as follows:
$$k = 1 \Longrightarrow \{(\sigma(q^k) < n) \vee (\sigma(n) < q^k)\}.$$

If $\sigma(q^k) < n$ and $\sigma(n) < q^k$ are both true, then the following lemma is violated:

Lemma 3 [Dris - 2013 (currently a work in progress)]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  Then
$$\frac{\sigma(q^k)}{n} + \frac{\sigma(n)}{q^k} > 2\sqrt[4]{\frac{8}{5}}.$$

Thus, we have the following result:

Theorem 1 [Dris - 2013 (currently a work in progress)]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  If $k = \nu_{q}(N) = 1$, then either $\sigma(q^k) < n$ or $\sigma(n) < q^k$, but not both.

However, we have the following further results by the author:

Lemma 4 [Dris - 2012]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  If $n < q$, then $k = \nu_{q}(N)= 1$.

Additionally, we have:

Lemma 5 [Dris - 2012]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  If $$\frac{\sigma(n)}{q^k} < \frac{\sigma(q^k)}{n},$$
then $n < q^k$.

Lastly, the following lemma is trivial:

Lemma 6 [Dris - 2013 (currently a work in progress)]  Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  If $n < q^k$, then $k = \nu_{q}(N)= 1$ if and only if $n < q$.


Together - Theorem 1, Lemma 4, Lemma 5 and Lemma 6 imply the following result:

Theorem 2 [Dris - 2012]   Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  Then the following biconditional is true:
$k = \nu_{q}(N)= 1$ if and only if $n < \sigma(n) < q^k < \sigma(q^k)$.

Proof:
From Lemma 2, if $k = \nu_{q}(N) = 1$ then the only possible scenarios are the following three (3) possibilities:

$$q^k < \sigma(q^k) < n < \sigma(n)$$
$$n < q^k < \sigma(q^k) < \sigma(n)$$
$$n < \sigma(n) < q^k < \sigma(q^k).$$

We claim that $q^k < n$ if and only if $\sigma(q^k) < \sigma(n)$, if $k = 1$.  It suffices to dispose of the second case.  To this end, by Theorem 1, since $k = \nu_{q}(N) = 1$, we get:

Either $\sigma(q^k) < n$ or $\sigma(n) < q^k$, but not both.

In the first case, we have:
$$q^k < \sigma(q^k) < n < \sigma(n)$$

In the second case, we have:
$$n < \sigma(n) < q^k < \sigma(q^k)$$

(Note that $k = \nu_{q}(N) = 1$ serves as the value of the "place holder" for the exponent of the Euler prime $q$.)

By Lemma 5, the second case implies $n < q^k$, which in turn implies the biconditional $k = 1$ if and only if $n < q$.  Additionally, since our underlying assumption is $k = \nu_{q}(N) = 1$, we "substitute $1$ for $k$ (everywhere it occurs)", and thereby have:

$k = 1 \wedge n < \sigma(n) < q^k < \sigma(q^k) \Longleftrightarrow \frac{\sigma(n)}{q} < 1 < \frac{\sigma(q)}{n} \Longleftrightarrow n < q \Longrightarrow k = 1$

Consequently:
$k = \nu_{q}(N)= 1$ if and only if $n < \sigma(n) < q^k < \sigma(q^k),$
and our claim in this particular proposition is proved.

By the author's previous claims in [Dris - 2012] (which is an earlier version of [Dris - 2012], we have the following corollaries.

Corollary:
Let $N = {q^k}{n^2}$ be an OPN given in Eulerian form.  Then the following propositions are true:
(a)  If Sorli's conjecture is true, then the Euler prime $q$ is the largest prime factor of the OPN $N$.
(b)  If Sorli's conjecture is true, then $q > \sqrt[3]{N}$.  In particular, $q > {10}^{500}$ [Ochem, Rao - 2012].
(c)  If Sorli's conjecture is false, then $\sigma(q^k) < n$.

Remark 1:  In other words, Sorli's conjecture is true if and only if $\sigma(n) < q^k$.  Another way of saying this is that Sorli's conjecture is false if and only if $\sigma(q^k) < n$.

Remark 2:  From Remark 1, we then deduce that Conjecture 1 is equivalent to the statement that Sorli's conjecture is false.

Author's final notes:  I am in the process of coming up with a pre-final revision to my earlier submission(s) to arXiv, per an e-mail advice from the moderators regarding my third paper installment (i.e., http://www.scribd.com/doc/132988214).  I will probably be able to upload the revision(s) to arXiv by April 10 at the latest.  (Edit: [April 15 2013 - 9:00 PM Manila time] - The third revision to http://arxiv.org/abs/1302.5991 is set to appear by tomorrow morning.  In particular, the proof of the Conjecture alluded to in the final section contained therein rests on ruling out the possibility that $k = 1$ might imply $\sigma(q^k) < n$.)

Addendum:  Recently, the inequality $n < q$ was almost disproved in [Acquaah, Konyagin - 2012].  Acquaah and Konyagin obtain $q < n{\sqrt{3}}$, if Sorli's conjecture is true.

WWW Hyperlink: An introduction to the study of perfect numbers and related topics

There is a page in the Arizona Math department website that features "A Study of Perfect Numbers and Related Topics, With Special Emphasis on the Search for an Odd Perfect Number [OPN]" by J. W. Gaberdiel.  Anyone who would wish to do research on the topic of OPNs should check this out...

My UPOU - DCS Essay (Circa 2009)

I prepared a short essay [in early 2009] consisting of three paragraphs (and composed of less than 500 words) in compliance with a requirement in an application  for admission to the Distance Education (DE) program of Diploma in Computer Science (DCS) in the University of the Philippines (UP) - Open University [UPOU].

I still have a hard copy of the said essay, so I thought it might be helpful for prospective UPOU - DCS applicants to check it out.  There is an online copy in Scribd.

Please do check out Google for more information about the "UPOU - DCS" DE program.

OPN Research - February to March 2013

Let $N = {q^k}{n^2}$ be an odd perfect number (OPN) given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$).  Note that $q \neq n$ and $q^k \neq n$.

In his Ph.D. thesis completed in the year 2003, Ronald Maurice Sorli conjectured that $k = 1$.

In the author's paper titled "The Abundancy Index of Divisors of Odd Perfect Numbers" published in the Journal of Integer Sequences in September 2012, he proved that $n < q$ implies $k = 1$.  He likewise conjectured that $q^k < n$, on the basis of the related inequality $I(q^k) < \sqrt[3]{2} < I(n)$, where $I(x) = \frac{\sigma(x)}{x}$ is the abundancy index of the (positive) integer $x$.

It is now natural to ask as to whether the converse "$k = 1$ implies $n < q$" holds.  It is easy to see that this is true if $n < q^k$.  From earlier blog posts here, we have shown that if $q^k < n$, then $k \geq 1$ if and only if $q < n$.  Consequently, to prove the biconditional "$k = 1$ if and only if $n < q$", it suffices to rule out the conjunction $q^k < n$ and $k = 1$.

Update (February 25, 2013) - I have decided instead to upload a paper containing the details for some new results that I have obtained related to Sorli's conjecture.  The paper is set to appear in the arXiv in a few days.

Update (February 26, 2013 - 9:00 AM MNL time) - The paper has just appeared in the arXiv.

Update (March 27, 2013 - 10:30 PM MNL time) - A sequel to the first paper uploaded last February 26 is now available in the arXiv.

Update (March 29, 2013 - 12:15 AM MNL time) - The third installment to this series of papers for Feb-Mar 2013 will appear in the arXiv at around 9:00 AM MNL time today.

Update (March 30, 2013 - 11:30 AM MNL time) - There is some delay in the posting of the third installment to arXiv.  I have temporarily uploaded a copy to Scribd.

7 nutrients all men need

Source: http://ph.she.yahoo.com/7-nutrients-men-need-085754312.html

Selenium

Selenium is a powerful antioxidant which can help to reduce or prevent hair loss in men. Furthermore, studies have suggested that selenium can boost sperm health and motility, improving fertility. Selenium is also great for lowering “bad” cholesterol, preventing blood clots and lifting your mood. To get your recommended intake of selenium, try snacking on Brazil nuts, which are one of the most concentrated sources of the mineral, or eating more fish and seafood.

B vitamins

B vitamins are essential for general wellbeing and can help to alleviate depression, promote a healthy nervous system and boost energy levels. Studies have also shown that folate (vitamin B9) can help to keep sperm healthy, while biotin (vitamin B7) can help to treat hair loss. Good sources of B vitamins include whole grains, legumes, nutritional yeast and green leafy vegetables.

Zinc

Zinc is essential for men’s fertility and sexual health. The mineral not only helps to maintain healthy testosterone levels and boost libido, it is also essential for healthy sperm production. One of the best sources of zinc is oysters, although pumpkin seeds, meat, oats and other shellfish are all good sources.

Omega-3 fatty acids

Omega-3 fatty acids are one of the most multi-purpose nutrients around and can help to address many of men’s most common health complaints. Omega-3 fatty acids have been linked to lowered levels of "bad" cholesterol and can also reduce risk of many illnesses, including heart disease, colorectal cancer, prostate cancer and depression.  One of the best sources of omega-3 fatty acids is oily fish such as salmon, mackerel and sardines. 

Vitamin D

As well as helping to promote good bone health and prevent risk of mental illness and heart disease, according to a study led by a researcher at the Harvard School of Public Health, high levels of vitamin D can lower a man’s risk of dying from prostate cancer. Studies have also suggested that vitamin D can help to improve men’s sex drive by boosting testosterone levels, with research suggesting that an hour of sunshine can boost a man's testosterone levels by 69 per cent. Good food sources of vitamin D include oily fish and egg yolk.

Lycopene

Studies have found that lycopene – the carotenoid that gives tomatoes their color – can help to reduce risk of colorectal cancer, lower cholesterol and reduce the risk of heart disease; the leading cause of death in men. Research has also shown that men who frequently eat foods rich in lycopene may drastically reduce their risk of developing prostate cancer and that lycopene can slow the growth of, or even kill, prostate cancer cells.

Magnesium

Magnesium is important for healthy bones, energy levels and muscle function, as well as many other parts of the body and other essential functions. Furthermore, research has suggested that getting enough magnesium can help to reduce men’s risk of colon cancer. Good sources of magnesium include leafy green vegetables, nuts and seeds.

End Near? Doomsday Clock Holds at 5 'Til Midnight

Source: http://ph.news.yahoo.com/end-near-doomsday-clock-holds-5-til-midnight-232147095.html

The hands of the infamous "Doomsday Clock" will remain firmly in their place at five minutes to midnight — symbolizing humans' destruction — for the year 2013, scientists announced today (Jan. 14).

Keeping their outlook for the future of humanity quite dim, the group of scientists also wrote an open letter to President Barack Obama, urging him to partner with other global leaders to act on climate change.

The clock is a symbol of the threat of humanity's imminent destruction from nuclear or biological weapons, climate change and other human-caused disasters. In making their deliberations about how to update the clock's time this year, the Bulletin of the Atomic Scientists considered the current state of nuclear arsenals around the globe, the slow and costly recovery from events like Fukushima nuclear meltdown, and extreme weather events that fit in with a pattern of global warming.

"2012 was the hottest year on record in the contiguous United States, marked by devastating drought and brutal storms," the letter says. "These extreme events are exactly what climate models predict for an atmosphere laden with greenhouse gases." [Doom and Gloom: 10 Post-Apocalyptic Worlds]

At the same time, the letter did give a nod to some progress, applauding the president for taking steps to "nudge the country along a more rational energy path," with his support for wind and other renewable energy sources.

"We have as much hope for Obama's second term in office as we did in 2010, when we moved back the hand of the Clock after his first year in office," Robert Socolow, chair of the board that determines the clock's position, said in a statement. "This is the year for U.S. leadership in slowing climate change and setting a path toward a world without nuclear weapons."

The Doomsday Clock came into being in 1947 as a way for atomic scientists to warn the world of the dangers of nuclear weapons. That year, the Bulletin set the time at seven minutes to midnight, with midnight symbolizing humanity's destruction. By 1949, it was at three minutes to midnight as the relationship between the United States and the Soviet Union deteriorated. In 1953, after the first test of the hydrogen bomb,the doomsday clock ticked to two minutes until midnight.

The Bulletin was at its most optimistic in 1991, when the Cold War thawed and the United States and Russia began cutting their arsenals. That year, the clock was set at 17 minutes to midnight.

From then until 2010, however, it was a gradual creep back toward destruction, as hopes of total nuclear disarmament vanished and threats of nuclear terrorism and climate change reared their heads. In 2010, the Bulletin found some hope in arms reduction treaties and international climate talks and bumped the minute hand of the Doomsday Clock back to six minutes from midnight from its previous post at five to midnight. But by 2012, the clock was pushed forward another minute.

6 Myths About Sleep Busted

Here are 6 sleep myths to ignore, as reported in EatingWell Magazine.

Myth: Falling asleep to the TV is OK.
The Truth: Artificial light from televisions-and especially from computer and smartphone screens-may suppress production of melatonin, a sleep-inducing hormone triggered by darkness. Artificial light also shifts your circadian rhythms-a biological cycle that responds primarily to daylight and darkness and influences sleep. 

Myth: A glass of wine before bed will help you get a better night's rest.
The Truth: Because alcohol is a sedative, drinking wine, beer or other alcoholic beverages may help you fall asleep, but as little as two drinks can cause you to sleep less restfully and wake up more frequently. And alcohol-related sleep disturbances are worse for women, say researchers at the University of Michigan. Drink moderately, if at all, and avoid drinking within a few hours of bedtime.

Myth: Exercising at night keeps you awake.
The Truth: Hitting the gym or going for a run less than 3 hours before bedtime won't prevent you from falling asleep, according to recent research. It may, however, hinder your sleep quality.

Myth: A cup of herbal tea will put you to sleep faster.
The Truth: Though chamomile, lemon balm, hops and passionflower are all touted for their sleep-promoting properties (and are often found in "sleep-formula" tea blends), their effectiveness hasn't been proven in clinical studies, according to the American Academy of Sleep Medicine. 

Myth: You can catch up on lost sleep by sleeping in on weekends.
The Truth: If you sleep poorly-or don't get enough sleep-once or twice a week, you can make up for it. But after more than a few sleepless nights, it becomes harder to "recover" from lost sleep, says new research from Penn State.

Myth: Drinking a glass of warm milk will help you fall asleep.
The Truth: The theory is this: milk contains tryptophan (the amino acid best known for being in turkey), which when released into the brain produces serotonin-a serenity-boosting neurotransmitter. But when milk was tested, it failed to affect sleep patterns. "Tryptophan-containing foods don't produce the hypnotic effects pure tryptophan does, because other amino acids in those foods compete to get into the brain," explains Art Spielman, M.D., an insomnia expert and professor of psychology at the City University of New York.

Source: http://ph.she.yahoo.com/6-myths-about-sleep-busted.html