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

Amazing Story January 28, 2017



The Amazing Story


This is an amazing story. It begins back in 1990 and continues even as we speak.

It takes about 26 minutes to hear/see the whole story but I think you’ll find it’s time well-spent if you can give it your attention; especially without skipping parts.

This is the story of the century. It makes WoodStein look like amateurs. Or…  Anyway, 26 minutes is a tiny price to pay for what may be the story that will be talked about for hundreds of years; or it may be… In any case I find it more engrossing than any Harry Palmer adventure. It starts out a bit slow but builds steadily until, by the end, we say to ourselves

“HOLY SHIT! How come this isn’t the lead story

on every newscast every day all day?”


But seriously… how come? See what you think.

Due to GoDaddy file size limitations, the story had to be broken up into 5 videos of about 4-5 minutes each.

Btw, who should play the part of Christopher Steele?


Part 1 of 5 below.


Part 2 of 5 below


Part 3 of 5 below

Part 4 of 5 below



Part 5 of 5 below


Holy Shit!

Short Summary below

An amazing story… right?






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Brilliant, Wacko, or Super Drama Queen January 17, 2017




 There’s something about her that keeps me coming back. 

I can’t decide whether:








  • She’s the most brilliant detective and investigative reporter since Woodstein (see below), or
  • She’s a wacko conspiracy theorist with a flair for the dramatic, or
  • She’s a drama queen/actress par excellence with a flair for connecting conspiracy dots (a life and death story), or,  not acting and possibly headed for a  meltdown.


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Whatever it is I keep coming back; maybe not for the same reasons as many others but I come back nonetheless.  Anyway, here is one of the most entertaining fulminations I have ever witnessed.

“… A life and death story… “

Hang in there Rachel.  It’ll all be ok.  I promise.


Open The Gates!






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Pay no attention to the man behind the curtain January 5, 2017

You can’t make this stuff up. Or maybe they can. Or maybe they did. Wait a minute!… I remember now… They did! Am I the only one who connects dots that they didn’t intend to be connected (or even noticed)?

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Last night (1/4/16) on MSNBC they brought in a fellow named Paul Farhi, of the Washington Post, to comment on and reinforce the host’s agenda regarding FAKE NEWS. The host, and Paul Farhi (of the Washington Post), were speaking on how horrifying “FAKE news” is



Did I mention that Paul Farhi is from the Washington Post?; the same Washington Post that was caught red-handed publishing the FAKE NEWS about “Russian hackers” penetrating the U.S. electrical grid via a Vermont electric utility. I love that word “penetrating.” And that was less than a week ago! But the MSNBC host AND Mr. Farhi seem to have forgotten that fact. All that said, here’s the shortened video from MSNBC.

You can find the entire video here:


And if you don’t know about, or want to be reminded about, the Washington Post’s FAKE NEWS story of 12/31/16 then just read the following posts.

Fake News

Fake News Follow Up

Russian Porn In Vermont

Fake Fake Fake Fake


It’s a rather dramatic fall for an institution like the Washington Post to go from All The President’s Men to Fake,Fake,Fake,Fake (and then to pretend that nothing happened).  And for Paul Farhi, now an errand boy sent by grocery clerks.


Open The Gates!




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Fake Fake Fake Fake January 4, 2017



The Russians Are Coming… Again!

First the Washington Post reported that Russian hackers had penetrated the U.S. electric grid.   Then NBC got hold of the story and created a top-notch video which you just have to see. It’s really quite dramatic!!! (at the bottom of this post).


Then the Washington Post retracts the story, now saying that “Russian operation hacked a Vermont utility, showing risk to U.S. electrical grid security, officials say”  Note the change from penetration to “showing risk”


Then we keep reading and we find out all kinds of interesting things… until we are left to reasonably conclude that it was all FAKE.

If you have a mind to, read just some of what was found and judge for yourself (reprinted here as PDF files).








The Washington Post, NBC, and others… “fake fake fake fake.”


Here is NBC’s video version of the Washington Post story.  The drama… the drama.










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Russian Porn In Vermont January 2, 2017


The Drama… the drama.

What REALLY Happened…

So let’s say you’re some red blooded and horny American guy at the beginning of your career. You can’t help yourself. You may even have been at work when… you know… you make a visit… you know… to that porn SITE… you know… the one with Anastasiya and Svetlana.

Or you may have been at home when you visited You thought it would be fun AND SAFE to put those kartinki on a thumb drive and take them to the office.  And you know, impress the boss by pretending to work, while… you know… with… you know… Svetlana.


Who would have thunk it? How were you supposed to know that Rooskie hacking shit you hear so much about would literally jump from your browser at work, on your lunch hour, into the power grid!? All you did was click on the phrase, in HUGE bold print, that said, “Click Here to see Anastasiya make Svetlana happier than Vlad on a horse.”


Or who knew that plugging that fucking “BestOfMoscow” thumb drive into your cubicle laptop would, somehow, fuck up your office computer and eventually take out power for all of Vermont!

Holy Shit!

But as Bob Dylan said…  God is on our side! The boss knew what to do when the shit hit the fan! He took the thumb drive from you and flushed it down the toilet in the Men’s Room. Then he picked up the phone and called the FBI and screamed “The Russians!… The Russians… They’re here!” Then the boss took you to McDonalds (for the wifi) where the two of you created a Yahoo! email account and sent NBC an anonymous letter on some real old piece of shit laptop. When it was all over the boss said “Throw this laptop in the dumpster at your apartment building. And, you owe me one. And it will be a big one.”


You owe him one… You owe him one… And what did he mean by the term “big one?”…. hmmm….. uh oh.

The drama… the drama.




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One Thing Leads To Another December 26, 2016

one thing leads to another…


The Prime Factor Calculator Project in Csharp


Insights Into Internet Security



For ease of writing, I’ll call the Prime Factor Calculator, the PFC.

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The Original Plan – Build a personal Class Library


The Prime Factor Calculator app (in C#) is an unintended byproduct of a recent “Class Library” project I started. My development projects are typically one-off programs and because of that I’d rarely take the time to put functionality into separate thoughtfully designed classes (let alone in a class library!). In general I would just code the routines I need as just additional functions in the overall program I’m working on. If I happen to need that functionality in a later project I’d just cut and paste code from the earlier project into the program(s) of the new project (and modify as needed). Yeah, sloppy. I know. But lately I have been wanting to use some earlier code in all new projects (in new programs). So, I decided to develop a Class Library with cleaner and more thoughtful versions of the functionality I want in it. And then to go back and clean up the earlier projects (making use of the new Class library).


In the new Class Library I decided I wanted (needed?) a class to compute Prime Factors for numbers that you pass to it. Although a Prime Factor computation class is conceptually simple, in practice it takes a fair amount of research to figure out how to actually accomplish the task. And, if you develop a class to compute Prime Factors you will obviously need something to test it with!! The obvious thing to do is to develop a Prime Factor Calculator app (PFC).


The way I approached the problem was to implement the prime factor computation code in a class contained within the Prime Factor Calculator app.

.The Prime Factor Calculator app is developed in the Visual Studio IDE as a C# “WinForms” Desktop application… just in case you were wondering.


After the app is developed and tested I’d move the prime factor computation class to my new personal Class Library and delete it from the app itself. Then we test everything again. This is a simple and effective way to develop and test new classes.


In some sense we can think of the Prime Factor Calculator app (PFC) as just a test-driver for the computation class; but in reality it is much more than that. It is a simple yet useful app if you happen to have a need for computing the Prime Factors of numbers. Of course, there are more than a few PF calculator apps available “on the internet.” But so what… that’s like telling a car enthusiast that there’s a lot of ’57 Chevys out there so why bother restoring one yourself. Anyway, just below is a short video that shows what this particular PFC app is and what it does.


You might find it interesting to understand the basic techniques that were used in the PF computation class that actually does the factorization. It apparently came from somebody named Vishwas Garg. It took me a little while to comprehend and appreciate how it works. It’s really quite ingenious and reputed to be pretty efficient. Anyway, you can find a link for, and a reprint, of his concept (pseudo-code) near the end of this post. All I did was implement it in C# using the MS .Net BigInteger class.


BTW, there are MANY Vishwas Gargs out there so I don’t know which Vishwas it was who proposed (created?) the factorization method I used.The Background is Done… Continuing the story…

While playing with and testing the PFC app it quickly became apparent that the amount of time needed to factor numbers increased non-linearly with the number of digits in the number.


Keep in mind that the PFC will show us how many milliseconds it takes for it to factor the numbers we give it.This non-linearity was not surprising nor should it be for anyone who has studied, even a little bit, prime factorization or cryptography or who has an appreciation for exponential growth functions. That said, I thought it worthwhile to actually graph some results. Here is one such graph (of PFC results only).


As we can see, it’s a pretty “classic” looking graph. So I starting thinking about it and I suspected that the graph and its data should be able to be closely approximated with a simple exponential formula.

Thinking back about the many texts/articles one reads, especially texts on cryptography, they will often say that the effort to factor a number will increase with the square of the value of the number. E.g. a 4 digit number will take 10 times more cpu cycles than a 3 digit number. But when I ran that theory (presumption) against the actual PFC results it did NOT hold water!   However, when using a spreadsheet to analyze the actual data it instead looked like the increase in time was roughly exponential based on 3.2 times the increase in the number of digits.   For example, an 11 digit number required 3.2 times more cpu cycles to factor than a 10 digit number.  And a 12 digit number required 3.2 times more cpu cycles than the 11 digit number;  and so on. 


Then there was that “aha” moment. 3.2 is roughly the square-root of 10!  This makes perfect sense because, when factoring numbers…

The square-root of the number being factored is as high as we need to go in our calculationsThat is because a Prime Factor of a number (any number) can NEVER be greater than the square-root of the number being factored!

And the factoring routine I was using accounted for that in its code… and that directly affects the cpu cycles needed to factor numbers!


So the next thing to do is to add this “theory” (formula) to the spreadsheet and produce a new graph!

Look what happened!



As we can see, the theoretical values (red line) based on the “3.2 conjecture” very closely matches the empirical reality (blue line based on actual observations)!

The Relationship To Cryptography


So now we come back around to cryptography and the internet. As you probably know, on the internet it’s all about public key cryptography (think SSL and https); and public key cryptography is all based on the difficulty of factoring large numbers.

So with this PFC app it takes about 500 seconds on my personal computer to factor 21 digit numbers. If one was depending on encryption using 21 digit numbers it would not be difficult to “crack” the codes. It would take less than 10 minutes. But let’s extrapolate a little…

How about encryption based on 100 digit numbers? That would require about

500 x 3.1679 = 1.49 x 1042seconds = ((1.49 x 1042) / 86400) days… or…

about 17,000,000,000,000,000,000,000,000,000,000,000,000 days!

But we might say, “Gerry… your computer is slow and your code is inefficient. Someone who writes really  efficient code with a FAST computer could do better.”   This is true.  However 


Let’s assume some other computer is a trillion times faster and someone else’s code is a billion times more efficient.  That still would require 17,000,000,000,000,000 days to crack the code.


. And even if, by some huge miracle you could shave off another 15 zeroes, all that’s needed is to increase the digit count from 100 to just 110 and we’re back up to over 2,700 years to crack the code by factoring the number! As we can see, the “Encryptors” will have the advantage over the “hackers” for Zillions of years unless some miracle of discovery happens in the field of Number Theory.


Now all of this we already knew at some level but by developing our own Prime Factor Calculator, making some observations, creating some spreadsheets and charts, and doing just a little analysis, we are able to get an excellent appreciation for public key cryptography which is based on factoring large numbers and which underpins the internet we know today.

So it seems reasonable to say that the state of public key cryptography is quite safe unless someone comes up with an unbelievably efficient way to factor numbers. In other words, it would require more than just faster machines and more efficient code! It would require a major breakthru in number theory! But I dare say such a discovery would be more notable than Einstein’s theory of relativity.

None of the above is really “new” but now I’ve had the opportunity to gain a better appreciation by “touching” and “feeling” it by coding, testing, and analyzing it in the real world; in my loft; in front of my $600 computer. 


So do I get a Starbuck’s gift card for all that?

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The following represents, roughly, the technique used by the “PF computation class.”   Actually I created 2 versions of this “code.” One is for factoring 64 bit unsigned integers (type ulong in C#). The other is for factoring HUGE numbers of arbitrary length via the BigInteger class which is part of Microsoft’s .Net class library.

One significant improvement I made to the routine is to move the sqrt calculation outside of the loops so that it’s only calculated once (it really only needs to be calculated once at the beginning of the function (and outside of the loops) as detailed earlier).

You may find it interesting to learn that, oddly enough, the BigInteger class does NOT have a method for obtaining square roots!  This means that we have to code our own square root routine which means we must code a binary search in order to get the square root of the numbers we want to factor!   One thing leads to another… or… It’s always somethin’  So… when you see   sqrt(n)  in the Vishwas’s code below, that’s a whole other routine you must code if you want this stuff to work for big numbers!  ooh!  ooh!  That could be another class to code and put in my class library!  This shit never fucking ends!



I found the routine at

but there are other copies/versions (e.g. pseudo-code) that are easily found.




// A function to print all prime factors of a given n


void primeFactors(int n)


// Print the number of 2s that divide n

while (n%2 == 0)


printf(“%d “, 2);

n = n/2;


// n must be odd at this point.

// So we can skip one element (Note i = i +2)


for (int i = 3; i <= sqrt(n); i = i+2)


// While i divides n, print i and divide n

while (n%i == 0)


printf(“%d “, i);

n = n/i;



// This condition is to handle the case

// when n is a prime number greater than 2


if (n > 2)

printf (“%d “, n);


/* Driver program to test above function */

int main()


int n = 315;


return 0;


Run on IDE


Output: 3 3 5 7


How does this work?

The steps 1 and 2 take care of composite numbers and step 3 takes care of prime numbers. To prove that the complete algorithm works, we need to prove that steps 1 and 2 actually take care of composite numbers. This is clear that step 1 takes care of even numbers. And after step 1, all remaining prime factor must be odd (difference of two prime factors must be at least 2), this explains why i is incremented by 2.


Now the main part is, the loop runs till square root of n not till. To prove that this optimization works, let us consider the following property of composite numbers.


Every composite number has at least one prime factor less than or equal to square root of itself.


This property can be proved using counter statement. Let a and b be two factors of n such that a*b = n. If both are greater than √n, then a.b > √n, * √n, which contradicts the expression “a * b = n”.


In step 2 of the above algorithm, we run a loop and do following in loop

a) Find the least prime factor i (must be less than √n,)

b) Remove all occurrences i from n by repeatedly dividing n by i.

c) Repeat steps a and b for divided n and i = i + 2. The steps a and b are repeated till n becomes either 1 or a prime number.


Thanks to Vishwas Garg for suggesting the above algorithm. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



The End















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Fake News Followup December 20, 2016

Fake News Followup

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When I published the Fake News post I sent an email to some folks letting them know about it. The email was short…

Subject: Fake News – And now for the rest of the story…

Roger Staubach goes after Trump over insult!

Extra! Extra! Read all about it!

Fake News | It Begs The QuestionIt Begs The Question




A few days later I received an interesting  response from one of the recipients:

Subject: Re: Fake News – And now for the rest of the story…

you actually used Fox News are [sic] reference for truth…

whats the matter the National Enquirer or The Onion didn’t cover the story.



My reply was…

Subject: Re: Fake News – And now for the rest of the story…

Fox was not the reference for truth. However, it indeed was the Fox News video that got me to actually READ the actual (so-called) “article” in order to actually find out the truth. But hey… who actually reads more than a headline anymore? Anyway, all that said, as a reference for the truth we can use the actual article.   It only takes a couple of minutes to read it. Anywho… I’m stumped… No matter how many times I read it I just can’t find where Staubach “goes after” Trump over an “insult.” I know the headline says it’s so but whenever I read it I just can’t find it. Very perplexing.

Roger Staubach goes at Donald Trump over Army-Navy game insult



Some interesting aspects of this Fake News followup are:

1-  Apparently the use of Fox News bothered him/her. But it’s my observation that this is true for MANY people. And that it’s all part of the “identity politics” phenomenon which I think more accurately is about “tribalism.” For many people (who identify as “Democrats?”) Fox News equates to Republicans which equates to tribal threat blah blah blah. Likewise, CNN/MSNBC equates to Democrats which equates to a threat to… Anyway, that’s my theory.

You may want to read Tribe by Sebastian Junger.   Or, there are 2 good videos (interviews) on Follow the link below.



2-   I was surprised that he/she apparently did not actually read the CBS Fake News “article” even though it was very short and prominently linked to in my own Fake News post. I (wrongly) assumed that a reader of my Fake News post would actually also read the actual CBS fake article; apparently a bad assumption. I’m guessing that the mere mention of Fox News was a showstopper for him/her.

So what’s the takeaway?

For many, only the headline matters?

We can’t be bothered with details no matter how easy to learn?

The “truth” is based on what tribal newsource we follow?

Open The Gates!

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Fake News December 16, 2016

It’s official (but we’ve known for a very long time). CBS and the New York Daily News are officially in the Fake News business.

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Here’s the Headline.

But before we get to the story, if you are one of the few that reads this blog,

After reading this post you will want to make sure to read the Follow up to this article






“Roger Staubach goes at Donald Trump

over Army-Navy game insult.”


Here’s the links to PDFs of the Headline and complete story as published by CBS and the New York Daily News.

The CBS Story    and    The Daily News Story


Here’s what really happened (click Play for the short video).


Somebody Is Turning Over In Their Grave

Make sure to read the Follow up to this article



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Recount November 30, 2016

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Today is Nov. 28, 2016. Last week Jill Stein    jillstein    told us that she would be asking for a recount for Wisconsin, Michigan, and Pennsylvania.

Then Hillary Clinton  brokehillary

chimed that her campaign would (i.e. she would) … From the NYTimes…

WASHINGTON — Nearly three weeks after Election Day, Hillary Clinton’s campaign said on Saturday that it would participate in a recount process in Wisconsin incited by a third-party candidate and would join any potential recounts in two other closely contested states, Pennsylvania and Michigan

The Clinton campaign held out little hope of success in any of the three states, and said it had seen no “actionable evidence” of vote hacking that might taint the results or otherwise provide new grounds for challenging Donald J. Trump’s victory. But it suggested it was going along with the recount effort to assure supporters that it was doing everything possible to verify that hacking by Russia or other irregularities had not affected the results.


My prediction is…

HRC and her campaign WILL, somewhere along the line (before Jan. 20?), claim that “those damned Rooskies hacked into some of the nation’s voting systems and we think we need to rerun the entire elections in at least those 3 states if not for all states because we just can’t let those damned Rooskies hack our computers and tell us who our President should be!. And furthermore…blah blah blah. AND I WANT MY RESET BUTTON BACK!!!resetbuttonOf course, a Clinton-friendly federal judge will readily be found to have the case put in front of so that he/she can agree, and issue a recount directive(s) in order to get the ball rolling (duh?).  

So that’s my prediction.

Interestingly, Clinton isn’t even pursuing the fact that Trump was not born in the U.S. and so is not even eligible to be President!!! He was born in… wait for it….. wait for it…..


You know.  I know you know!


Anywho…  there will be appeals and lawsuits by…  etc. etc. etc. …………… Until it reaches the Supreme Court. This could go on for a very long time unless SCOTUS decides otherwise. Soooo…..


This is where my prediction and its implications stop. It gets too fuzzy from here on. Too many odd things could then happen. For example…what happens if SCOTUS decides “sure, rerun the election(s).” Who would be president until the “3 state rerun” ends. And why just 3 states? Wouldn’t the issues be the same for all states??? And if it went in Clnton’s favor would Trump then try to get recounts in the other states? Could he claim that the Rooskies hacked the Recounts?  Would Obama take some sort of action(s) against the Rooskies? Would that become Obama’s new “Line in the sand?You may remember his last line in the sand. Anyway, will it lead to nuclear war? Would it lead to all-out CYBER WAR (sell AMZN now and buy guns, ammo, and cement!!!)? What happens if the “3 state rerun” went in Trump’s favor?… would HRC then try to find/claim other “Rooskie-hacked” states? You can see how complicated it gets.

Just a few last points not related to my prediction:

1. So what really are Stein’s motives? Perhaps you might want to read AND watch ALL of the following:

Yep. I believe her. Don’t you?

2. And just to reiterate from above:

The Clinton campaign held out little hope of success in any of the three states, and said it had seen no “actionable evidence” of vote hacking that might taint the results or otherwise provide new grounds for challenging Donald J. Trump’s victory. But it suggested it was going along with the recount effort to assure supporters that it was doing everything possible to verify that hacking by Russia or other irregularities had not affected the results.

Yep. OK. I can buy that… uh huh.

This blog posting is a time-capsule prediction. Time will tell… we’ll see what really happens.

One last point. You might think that I am against a recount, lawsuits, etc. The truth is that I am NOT against all that. In fact, I am in favor of it! It would make the most amazing and entertaining election and make the story even better! And I’m all for that! And… It makes the news much more interesting and entertaining.

Oh… btw, the “Manchuria” thing was a joke. 

Did you like the  Q♦  touch? 

Open The Gates!


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God Has OCD Part Deux November 8, 2016

God Has OCD – Part Deux

Did you know that a shortage of remainders is why the hypotenuse can’t have prime factors in common with the legs of a right triangle?  Interesting eh?  Want to know more?  Then read on.

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In part 1 of God Has OCD we dealt with the Pythagorean Theorem    A2 + B2 = C2

and we proved that, for primitive Pythagorean Triples, A and B do not ever “share” prime factors with C. That is, A and B do not ever have a prime factor that is also a prime factor of C.

In this article we’ll look into the “sharing” of prime fctors for the more general Sum Of Powers formulas of which the Pythagorean Theorem is merely a special case.

We can express the general Sum Of Powers formulas like so:

T1n + T2n + T3n .. + Txn = Zn

In the above we’ll call T1n + T2n + T3n .. + Txn the “terms” of the equation.

We’ll demonstrate how, for “primitive” N-tuples, where the greatest common denominator of all the terms is 1,  that the terms may or may not, “share” prime factors with Z; except for, again, the special case of the Pythagorean Theorem where prime factors are never “shared” and where there are only 2 terms (A and B as we’re accustomed to seeing them called).

So where do we begin? As I often do, I experimentally test theories first, and then see if they can be proven. In this case, I could again use my SumOfPowers program (in C#).

With the SumOfPowers program we can specify:

        • How many terms to use
        • A range of values for the terms
        • The exponent (n) to use



The program would then find integer solutions if any exist. Follow the link below for an example screen print of the SumOfPowers program after an execution.

SumOfPowers Example

For this article I ran the SumOfPowers program and the results demonstrated that sometimes the terms T1..Tx will have a prime factor in common with Z and sometimes none of the terms will have a prime factor in common with Z.

That said, we now unequivocally know and can demonstrate the answer with a real example but the question is why? And then, why is this NOT true for the case of 2 terms (e.g. the Pythagorean Theorem)?

Anyway here are a few simple examples found by the SumOfPowers program:

T13 + T23 + T33 .. + T63 = Z3

13 + 23 + 33 + 63 + 163 + 173 = 213

13 + 23 + 63 + 63 + 63 + 73 = 103

33 + 43 + 53 = 63

23 + 53 + 123 + 123 + 153 + 173 = 243

T15 + T25 + T35 .. + T85 = Z5

85 + 145 + 165 + 205 + 205 + 225 + 245 + 305 = 345



So why is it possible for the Sum Of Powers for more than 2 terms to share prime factors with Z while it is NOT possible when there are just 2 terms?

T1n + T2n .. + Txn = Zn            << YES prime factors sometimes shared!

T1n + T2n = Zn                     <<  NO.  Less than  3  terms then Prime factors NEVER shared!



The reason is as follows… If we divide both sides of the equation by one of Z’s prime factors we get something like this (for example):

33 + 43 + 53 = 63                               Prime factors of 6 are 3 and 2. Let’s use 3…

33      +      43      +      53      =        63

—-             ——           ——-            ——–

3                 3                3                 3



3 divides 3^3 and it also divides 6^3

however (4^3)/3 and (5^3)/3 are not integers but the equation is still valid because

The sum of all of the remainders is an integer multiple of 3. For example:

(4^3)/3 = 21 + 1/3 and

(5^3)/3 = 41 + 2/3 then

1/3 + 2/3 = 3/3 = integer

So it all works out because we still get integers on both sides of the equation.

But note that this could never work out when there’s only 2 terms because then there would only be a single remainder after dividing the terms T1 and T2 by one of the prime factors of Z. And with only a single remainder, the left side of the equation can’t possibly be an integer!

For example, for A2 + B2 = C2

let’s assume that one of the prime factors of C was also a prime factor of A. Let’s call that common prime factor Pc. Then we would have something like the following if we divide both sides of the equation by Pc:

(Pc x Pa2 … x Pan)  x A       +         B2        =         (Pc x Pc2 … x Pcn) x C

——————————                    —————               ————————————-

        Pc                                           Pc                                    Pc


Integer + ( B2 / Pc) = Integer

But    B2 / Pc        yields some remainder after division; And


Unlike the case of more than 2 terms on the left side of the equation,

there are no additional remainders possible to add to this remainder

in order to obtain an integer that would make the equation valid!!!


Integer   +    ( B2 / Pc)        =     Integer

Integer   +   NON-Integer    =     Integer            <<=== a contradiction



So our original assumption of some prime factor of C also being a prime factor of A can NOT ever be true for the Pythagorean Theorem or for any Sum Of Powers formula with just 2 terms on the left of the equals sign.



And in some quirky and curious sense

we can say that this is because,

for just 2 terms (like the Pythagorean Theorem),

there will always be a “shortage of remainders”

that would prevent the validity of the equation!!!

All done.  We can go now.

Open The Gates!

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