It Begs The Question
Hey…I'm just saying… And while we're at it, why are you defending them?

SumOf2Squares and Random Primes June 30, 2017


This post/article is a followup to a previous post about the Sum Of 2 Squares and how none of the terms of the formula can share prime factors. It’s required reading (but you’ll like it).


When last we left off, we had been discussing that:

1/2 of all primes are of the “SumOf2Squares” type (aka “SOTS” type”)


This ratio of 1/2 is seemingly extremely consistent

across all ranges of any significant size!


Anyway, so why are 1/2 of the primes seemingly of the “SumOf2Squares” type (aka “SOTS” type”)? Here is my conjecture (“explanation”). Follow me on this…


First of all, we know that a “SOTS prime” P is a prime that can be expressed as the sum of 2 squares; like so…

P = X2 + Y2And we also know that for this to be possible then (P – 1) must be divisible by 4. For example:

13 – 1 = 12 and 12 is divisible by 4 so 13 is a “SOTS prime” which means we can express 13 (P) as the sum of 2 squares; like so…

13 = 32 + 22

With that in mind, we might also observe that starting with 8, every other EVEN integer is divisible by 4. For example:

8, 12, 16, 20, 24, … are all divisible by 4.


From this we realize that every other ODD integer O will satisfy the first SOTS requirement of (O – 1) is divisible by 4

For example 9, 13, 17, 21, … 9-1 is divisible by 4, 13-1 is divisible by 4, and so on.


However, actually being a prime is the other/second requirement of being a “SOTS prime” and we can see that not every other ODD integer is a prime! For example, neither 9 nor 21 from the above list is a prime. But, what we’re after here is “explaining” why 1/2 of the primes are SOTS primes.


The Effect Of Random Distribution of Primes

Now, IF the primes are randomly distributed among the odd integers then we could expect as many primes P would randomly “land” on an ODD number that fullfilled the first requirement as would not. That is, it would be as likely as not that a prime P would “land” on an ODD number that met the requirement of (O – 1) being divisible by 4.   This would explain why, seemingly, and as conjectured in the prior post/article, that 1/2 of the primes are seemingly of the “Sum Of 2 Squares type.” That is, why 1/2 of the primes can be expressed as the Sum Of 2 Squares.

When I say “seemingly” it’s because it’s based on experimental evidence using the PrimeTest.exe program (see the prior post). Whew!.

It is generally thought that primes are “sort of” randomly distributed along the number line but within the “fact” that they “thin out” according to the Prime Number Theorem (PNT). Or put another way, within any significant/sizable range they are “kinda sort of pretty much” distributed randomly. We don’t know whether, or under what conditions, this “sort of nearly” random distribution falls apart. For example, does “nearly random” distribution of primes fall apart completely after 101234567890123456789? Or, conversely, could all primes greater than 101234567890123456789 be SOTS primes? Who knows. That said, at the bottom of this post are links to related articles that you may want to read.


So here is my first conjecture:

Due to the “pretty much” random distribution of primes then “pretty much” 1/2 of all primes will be SOTS primes..

The above first conjecture is pretty strong especially when using the term “pretty much.” It would probably (“pretty much”) not garner much support at the next AMS conference so let’s try this conjecture instead:


Due to the “random enough” distribution of primes even as we go to infinity, then we can say there are an infinite number of SOTS primes (an infinite number of primes that can be expressed as the Sum Of 2 Squares)..

I’m going to go with this last (second) conjecture and I’m sticking to it… but my gut tells me both are true.


The End…Except for the interesting links below.



There are many more articles to be found by searching on “random distribution of primes” or just “distribution of primes.”


New Pattern Found In Prime Numbers


Peculiar Pattern Found in “Random” Prime Numbers – Last digits of nearby primes have “anti-sameness” bias


Prime number theorem

Structure and randomness in the prime numbers


The End



No Comments on SumOf2Squares and Random Primes
Categories: Uncategorized

Gerry’s Sum Of 2 Squares Theorem June 28, 2017

Gerry’s YAPFO Theorem on 

Sum Of 2 Squares


After reading this article be sure to read the followup article explaining why 1/2 of all primes are of the “Sum of 2 Squares” variety.


.For whatever reason I seem to be fascinated with whether (or not)  the terms of various formulas have prime factors in common.  It is a subject of quite a few posts on this blog.  Anyway…


Fermat’s theorem on Sums Of 2 Squares states that an odd prime P is expressible as

P = X2 + Y2

with X and Y integers, if and only if   P  = 1 (mod 4)

That is, if P-1 is divisible by 4.

For example:

5 = 12 + 22,    41 = 42 + 52,    etc.

On the other hand, the primes 3, 7, 11, 19, etc. can not be expressed as the sum of 2 squares because

for them, P – 1 can not be evenly divided by 4.

Anyway, I recently stumbled into the above theorem via an excellent YouTube video on the Numberphile channel. Take a look (link below).

Does the guy in the following Numberphile video look like Max Von Sydow or what?



My initial thought was that the Sum Of 2 Squares theorem might make an interesting project with regard to which terms of the formula do (or do not) share prime factors (i.e. which terms might have prime factors in common). It would be another YAPFO project (YAPFO = Yet Another Prime Factor Oddity). Of course, I would start out by writing a program to show details for which candidate primes have terms with 1 or more prime factors in common.

P = X2 + Y2

Almost immediately it dawned on me that the “P” term could never share a prime factor with the other terms (X and Y) because P is prime, and X and Y are less than P.

The above immediately led to … Ok… so if P won’t be sharing a prime factor with either X or Y then the problem is reduced to whether or not X can share a prime factor with Y.

Ok. This could still be an interesting project with some coding. A couple of minutes later while making coffee it dawned on me that the X and Y terms could not possibly share any prime factors either. If they did share a prime factor we would end up with something like the following:

P = (Fc * Fx2 * Fx3.. * Fxn)2 + (Fc * Fy2 * Fy3.. * Fyn)2

where F? Are the prime factors of X and Y. And in particular, Fc is a prime factor common to both X and Y.

If there actually was a Fc common to both the X and Y terms then we can get the following…

P / Fc = ( Fx2 * Fx3.. * Fxn )2 + ( Fy2 * Fy3.. * Fyn )2

Non-Integer = Integer

which is a contradiction that tells us that there are no prime factors shared by both X and Y. So…Gerry’s theorem on



The Sums Of 2 Squares states that

for every odd prime P expressible as

P = X2 + Y2

There are not any prime factors common to

any of the terms for P, X, or Y.



So the whole episode was conceived as a YAPFO project but resolved itself in less than an hour so I guess we can’t really call it a “project.”   Oh well.   But it was still worth documenting.


Here are some related links you may want to follow:


The End

Breaking News!

It’s been 2 days since I first posted the above.   But for some reason I kept thinking about 

What portion of all primes could be expressed as

the sum of  2 squares (let’s call them “SOTS” primes):


P = X2 + Y2.

It should be rather simple to figure this out since I already had a program I’d written called PrimeTest.exe. I used that program for an article written almost 1 year ago called “Primality Testing For Huge Integers” that you can read about here:


For PrimeTest.exe we specify a starting number, and how many numbers to examine/test.    It will return summary stats about how many of the integers in that range are prime. And it will, if you want, also provide details about each of those primes in the range. It does other stuff too. Anyway, all I did was to modify the PrimeTest.exe code to:


.1.  Identify which primes in the selected range could be be expressed as the sum of 2 squares ( P = X2 + Y2 ). I.e. where (P – 1) is evenly divisible by 4.

.2.  At the end of each “test,” print stats on how many of the primes in the range were “SumOf2Squares” primes (SOTS primes).

.The testing results were extremely interesting although I have no explanation for them. As it turns out :


1/2 of the primes are of the “SumOf2Squares” type (“SOTS” type”)


This ratio is extremely consistent across all ranges of any significant size!


I found the above extremely interesting but also very odd. Based on experience, I was expecting the number of SOTS primes to vary in some way with the natural log of the count of all primes in general or to vary with the log of the size/value of a prime(s) .

Again, 1/2 of the primes are of the SOTS variety and this is extremely consistent and varies little!   But why this should be the case is a real mystery; and a very interesting one at that! Solving this mystery could make for a great project.


After reading this article be sure to read the followup article linked to just below:

why 1/2 of all primes are of the “Sum of 2 Squares” variety.


As a final note, below is an example of using the PrimeTest.exe program:



































No Comments on Gerry’s Sum Of 2 Squares Theorem
Categories: Uncategorized

Comey Was Raped June 12, 2017


James Comey is an imposing man. He’s 6’8″ tall with an athletic build. And he was the head of the FBI!!! – Arguably the most powerful law enforcement agency in the world and almost certainly in the U.S. Then he was raped by a 70 year old fat man when he was alone in a room with him. Well… maybe it was a figurative raping; but the feelings of inadequacy and shame were the same.

“I should have done more. I wish I had been stronger. I should have said ‘no.’  I shouldn’t have gone to the 2nd and 3rd meetings.”


Anyway… below are the contents of 2 opinion pieces recently published by the New York Times; they echo what I’ve said above (and then some).

Kudos to the NYT and the 2 brave sisters who connected the dots and shined a bright light on the real meaning of the recent Comey testimony. Read them below (or click on the link to the PDF reprint of the NYT web pages that you can find at the end of this article).


The Truth Will Set Us Free

And it’s in the New York Times OpEd section


First NYT Opinion Piece


As I listened to James B. Comey, the former F.B.I. director, tell the Senate Intelligence Committee about his personal meetings and phone calls with President Trump, I was reminded of something: the experience of a woman being harassed by her powerful, predatory boss. There was precisely that sinister air of coercion, of an employee helpless to avoid unsavory contact with an employer who is trying to grab what he wants.

After reading Mr. Comey’s earlier statement, I tweeted about this Wednesday night, and immediately heard from other women who had seen that narrative emerge. How recognizable it was that Mr. Comey was “stunned” to find himself in these potentially compromising positions. His incredulity, mixed with President Trump’s circling attempts to get his way, were poignant. For a woman who has spent a lifetime wrestling with situations where men have power they can abuse, this was disturbingly familiar.

On Jan. 27, Mr. Comey received a last-minute dinner invitation from the president, and then learned it would be “just the two of us.” On Thursday, Mr. Comey revealed that he had had to break a date with his wife in order to dine with Mr. Trump. Already, something about this “setup” made him “uneasy.”

The central business of this intimate dinner was Mr. Trump’s insistence: “I need loyalty, I expect loyalty.” Mr. Comey immediately recognized that this was a press for something he did not want to give. He froze: “I didn’t move, speak, or change my facial expression in any way during the awkward silence that followed.”

That reaction — the choice of stillness, responses calculated to neither encourage nor offend that characterized so many of his dealings with Mr. Trump — is so relatable for any woman. During his testimony, Mr. Comey was asked why he had not responded more robustly, why he had not told Mr. Trump that he, the president, was acting inappropriately or reported his behavior immediately to others in authority.

Mr. Comey expressed regret that he had not been “stronger” about it, but explained that it was all he could do to focus on not saying the wrong thing. In other words, he wanted to avoid granting any favor while avoiding the risk of direct confrontation — a problem so deeply resonant for women.

During that interminable, awkward dinner, Mr. Comey struggled to convince Mr. Trump of the danger of “blurring” boundaries. But Mr. Trump was not deterred and returned to the subject of the loyalty he must have. There you hear the eternal voice of the predatory seducer: the man who knows how hard he can make it for a woman to refuse his needs.

Mr. Comey tried to wriggle out of the trap being set for him. He offered his “honesty,” hoping this would appease his insatiable host. Mr. Trump countered with a demand for “honest loyalty.” Mr. Comey acquiesced. Yet as he documented this “very awkward conversation,” his concession of this phrase troubled him. He hoped he had not been misunderstood by the president.

The victim of sexual harassment is constantly haunted by the idea that she said or did something that gave her persecutor encouragement. Serial harassers, of course, have an intuitive sense of this, and are skilled at manipulating and exploiting it.

Mr. Comey, you are not alone. How many of us have played over and over in our minds an encounter that suddenly took a creepy, coercive turn? What did I say? Were my signals clear? Did I do something ambiguous? Did I say something compromising?

At a White House ceremony on Jan. 22, Mr. Comey reportedly tried to blend in with the curtains, so that he would not be noticed by the president. Mr. Trump called to him and pulled him, unwilling, into a hug. What woman has not tried to remain invisible from an unwelcome pursuer’s attentions?

To this series of bizarre interactions, in which he faced escalating pressure, Mr. Comey reacted with rising anxiety and distress. Time after time, Mr. Trump reverted to his questionable agenda, and Mr. Comey, at each pass, tried to parry the president’s unwanted advances.

This dynamic with the president became so disturbing to Mr. Comey that, after an Oval Office meeting in February, he implored the attorney general, Jeff Sessions, “to prevent any future direct communication between the president and me.” Mr. Comey did not want to be left alone with his boss again. We’ve been there, Jim.

In their final exchange, on April 11, Mr. Trump told the F.B.I. director, “I have been very loyal to you, very loyal; we had that thing you know.” On May 9, having rebuffed the president, Mr. Comey was fired.

“We had that thing.” Once more, the seducer asserts a shared intimacy that was not really there, attempting to ensnare his victim with an imputed complicity.

In the infamous “Access Hollywood” tape, Mr. Trump said of any woman he wanted: “I just start kissing them. It’s like a magnet. Just kiss. I don’t even wait. And when you’re a star, they let you do it. You can do anything.” And he added: “Grab ’em by the pussy. You can do anything.” With the power of the presidency at his disposal, Mr. Trump thought that he could use the psychology of coercive seduction on the nation’s chief law enforcement officer.

Victims of sexual harassment often face skepticism, doubts and accusations when they tell their story. That’s part of the predator’s power. But I’m here to tell James Comey, and all the women and men who have suffered at the hands of predators, I believe you



2nd Piece By NYT

Women Say to Comey: Welcome to Our World


A man is being publicly grilled about why he was alone in a room with someone he felt was threatening him. Why didn’t he simply resign if he felt uncomfortable with what his boss was asking him to do? Why did he keep taking calls from that boss, even if he thought they were inappropriate? Why didn’t he just come out and say he would not do what the boss was asking for?

Sound familiar? As dozens of people noted immediately on Twitter, if you switch genders, that is the experience of many women in sexual harassment cases. James Comey, the former director of the F.B.I., explained to senators during today’s hearing that he felt acutely uneasy and hesitant to directly confront his boss, the president of the United States. That’s right, even a savvy Washington insider, the same height as LeBron James and no stranger to the cut and thrust of power, seemed slightly ashamed that he had not been able to do so.

“Maybe if I were stronger, I would have,” he said, trying to answer a question about why he didn’t speak his mind. “I was so stunned by the conversation that I just took it in.”

These are the emotions that many women have struggled to explain in the face of sexual harassment, and the ones that have often given defense attorneys grist for what appear to be inconsistencies.

Imbalance of power often lies at the heart of sexual harassment or assault cases, from those of Roger Ailes and Bill O’Reilly at Fox News to the trial of Bill Cosby, underway the same day as the hearing of the Senate Intelligence Committee. On Wednesday, Andrea Constand, Mr. Cosby’s accuser, concluded two days on the witness stand, with defense attorneys suggesting that her continued contacts with Mr. Cosby undermined her credibility. Unsurprisingly enough, today’s hearing shows that power can discomfit and silence men as well as women.

Sexual harassment and assault often provoke debates about credibility, fairness and bias. But at least for today, the tables were turned, and men could glimpse what women have often endured.



Here are links to PDFs that are reprints of the NYT articles from their web site.



The End


















No Comments on Comey Was Raped
Categories: Uncategorized

Comey And The Drama The Drama June 11, 2017

The drama.   The drama.

Shortly after I first published this post I was discussing it with a friend who had read it.  The first thing he/she wanted to do, and what he/she repeatedly wanted to do, was to say… “But Trump this…”   or   “But Trump that…”   I had to continually remind him/her that

This posting is NOT about Trump.  

It seems that so many people want to conflate issues about anything even remotely political with issues about Trump.  I see it constantly and I am sure you do too.  The classic case is Rachel Maddow… but she’s into it on a much different level (Tin Foil Hat  level).  And she’s entertaining in a obsessive / compulsive / train wreck  sort of way  but still wildly entertaining.  I often watch her show and wish I could date her.  Then I could make the movie “My dinner with Rachel.”    But I digress… Let me repeat;

So many people want to conflate issues

about anything even remotely political

with issues about Trump.

To all who do that, STOP!  Just STOP!  Make the effort to develop other interests!  That said let’s continue…

Lock The Gates!


And remember… this posting is NOT ABOUT TRUMP

Now click play



“Former FBI director James Comey hinted Thursday at the identity of the source who provided to the media the content of his memos of conversations with President Donald Trump.”

Comey said his source, which he instructed to provide information on the memos to the media, was a Columbia law professor who was a close friend.


So why was he so coy? Why not just name him? Oh, I get it; because he makes himself look “loyal” to a good friend even though we all know, including Comey,  that with the info he gave out it would only take someone about 10 minutes (or less) to figure out who he was talking about. Does anyone really doubt this?

Daniel Richman was later revealed to be that professor. He confirmed that he was the source of the memos’ contents to The Huffington Post and other outlets.


So what was the deal with all of the Comey drama?

Why all the drama and the so-called “leaking” thru a “friend?”

Does this really constitute a “leak” or is it a charade?

Charade: an absurd pretense intended to create a pleasant or respectable appearance.

The drama…The drama.




Comey knew he was going to talk about all of this in a few days anyway! Why not just release his own memos himself?

No balls? Likes the spotlight? Likes the buildup to his appearance?

But he’s an upstanding guy with integrity.” Uh Huh… Sure… That’s why he dragged his “good friend” into the whole affair. Yeah… let’s go with that… upstanding and loyalty to a friend.


OK;  after you’ve thought about it for a while…

How much of the “Comey story”  do you buy?

How much of the “Comey story” did you buy last October?

Are we starting to see a pattern here?



Open The Gates!




No Comments on Comey And The Drama The Drama
Categories: Uncategorized

The Home Storage Gold IRA Solution June 6, 2017

Home Storage Gold IRA – What Was Weird Solution


Look at the photo above.  It’s from the very beginning of the commercial.  

How far did the DOW plummet?

How far could it possibly plummet?………….Now we understand the problem.



No Comments on The Home Storage Gold IRA Solution
Categories: Uncategorized

The Home Storage Gold IRA

So I’m watching the tube today and I see an ad from “The Capital Gold Group.” It’s about how you can have an IRA where you invest in gold and actually have the gold delivered to your home where you store it yourself (in a hole in the back yard?). Anyway, the beginning of the commercial went by VERY fast but somehow it really caught my eye. It made me ask “Did it really say that? Or did I imagine it?” It was so interesting and strange I had to pause the TV and get my camera/phone to take a photo of the TV screen.

Here are a couple of photos from the commercial. Can you guess what is so interesting/bizarre? No… I’m not going to divulge it just yet (the discovery process is supposed to be fun). Just study the photos and when you are ready you can click on the link at the bottom of this post to find out what is so strange; at least, what I thought was strange.



Anyway, the 2 links just below go to the Capital Gold Group’s web site (or you can google them and get there that way). The second link is more to the point and takes you right to their “StoreTheGoldAtHome IRA” product.


Now click here —>> For the answer why I thought their TV commercial was so weird.


The End





No Comments on The Home Storage Gold IRA
Categories: Uncategorized

Tin Foil Hat Area June 4, 2017



Shhhhhhh!   Let’s listen in….

Lock the Gates!


Did I just hear her say that it was actually Trump that told the Russians what to do?

That Trump told those damned Roosskies what to release and when?

And did I just hear that those damned Macedonians fucked her over too?

And they were also responsible for her losing… with their damned Macedonian

internet FakeNews sites that fooled us, the American voters.

Yup… it almost fooled me too… and now I’m pissed! Did it fool you?

If only Jeb had done his part and won the nomination! 

Then the world would have made sense again. It’s all Jeb’s fault!…

Those fucking Bush bastards…

I just don’t understand why others can’t see what I do…

Why they can’t connect these dots.

I just don’t understand.

I’m so depressed.

Open The Gates!



No Comments on Tin Foil Hat Area
Categories: Uncategorized

Build A Secure PC Environment June 1, 2017

This post is about how to build and use a very secure PC environment.  Not much to say here because for this article

I decided to put it all into a PDF document that you can access/read by

clicking here—>  SecurePCasPDF




No Comments on Build A Secure PC Environment
Categories: Uncategorized

Dan Rather – I Remember That Asswipe May 19, 2017

Dan Rather on Rachel Maddow…

And on other “news” shows where they’ve conveniently “forgotten”

why Dan Rather was fired from his job as the

anchor on the CBS Evening News.

So watch the video. Then… (see below)…



Maybe you are too young, or maybe you don’t remember, or maybe you never cared. But maybe, just maybe, you’re as amazed and outraged as I am that ANY of these “news” nets would EVER even give Dan Rather the time of day let alone put him in front of a camera; Because

I know why Dan Rather was fired from his job

as the anchor on the CBS Evening News.

No Comments on Dan Rather – I Remember That Asswipe
Categories: Uncategorized

Unintended Consequences May 14, 2017

Can you get wifi in the parking lot?

The NSA creates a hacking tool and what happens next?  You guessed it.  Hackers get hold of it which leads to…

Unintended Consequences

(At least we hope it was unintended)

The world is a fragile place; a very fragile place.

Entropy attacks in unexpected ways.

If they had just asked I’d have told them this was bound to happen.  No… seriously.  But hey… they don’t listen to me…. 

Although… once they stumble on this post maybe they’ll start.


Read this article — >>  GuardianNSAwannacryptor

And of course, you can find much more on the subject by using google.





No Comments on Unintended Consequences
Categories: Uncategorized