## JSH: Residues and factoring

I'm hoping to shorten the arguing phase with the remarkably simple solution to the factoring problem I've outlined in a previous post by pointing out a few things:

Given a target composite T, and

z^2 = y^2 + nT

where n is an integer chosen for z to have 3 as a factor—so if T mod 3 = 1 then n=5 will work—then z exists as

z = (f_1 + f_2)/2

when f_1*f_2 = nT, and of course you want integer factors.

The method I have then allows you to find z using a variable I call k, where

z = 3k/2

and k^2 = 2^{-1} (nT) modulo p

where p is an odd prime less than k as k is an even positive integer.

May seem complicated all in a rush like that but it's incredibly simple.

The technique boils down to getting the residue of z modulo an odd prime p, by using the help of a variable I call k, and it just so happens it's easy to do that, if you think of it, but I guess no one else did so the factoring problem was wrongly thought to be hard.

But you may wonder, how about multiple solutions for z?

Well, multiple k's would just have to have the same residue modulo p.

Ok, then you may ask, how do you pick p?

Answer is that you just try some odd prime less than the minimum k, where since

z = 3k/2, and z = (f_1 + f_2)/2

you can find the minimum possible k by assuming f_1 = f_2, which is the case of nT being a perfect square.

So you just try some odd prime p less than the minimum k, and see if it works such that k exists with

k^2 = 2^{-1} (nT) modulo p

and if it doesn't you just try another prime! And you know that 50% of the time a given prime p should work.

You may wonder, is there any doubt about the proof or any way to question whether it works?

Short answer is, no. It's trivial algebra. The big thing that happened here is the use of a massively powerful although simple idea.

It just so happened that there was this clever way to figure out the residue of z modulo chosen odd primes, where I call them helper primes as all they do is help you factor your target composite T.

They were always there, waiting to be used.

Sometimes in mathematics a solution can be about luck and looking for something simple.

If you think a problem is hard, sometimes you can make it so, while doing real mathematical research is about not assuming what is not proven.

Maybe the greatest lesson for many of you to learn in your mathematical careers.

Mathematical proof is what you rely on, not social things like how many mathematicians before you looked at a problem, as they could ALL have missed something simple.

## So I solved the factoring problem

Wow. Years of research and then the answer turns out to be incredibly simple, but that's how it can be with mathematics.

What I like about this method is that there isn't really any room for people to argue with me about whether or not it solves the factoring problem (I've posted it in another thread), and it has to be incredibly fast, as easily shown by mathematical proof.

So no dumb arguments about me having to demonstrate as I haven't gotten around to testing it yet, at all, as I've just relied on mathematical proof and it's so simple.

It boggles the mind how simple it is.

Needless to say this research result gives FULL VALIDATION to my research across the board and will usher in a new era in mathematics as a discipline around the world.

That brings forward the correct prime counting function, the correct information about randomness and primes, so that the Goldbach Conjecture is handled and the Twin Primes conjecture is handled, and the Riemann Hypothesis can probably be quickly handled.

Also then the correct short proof of Fermat's Last Theorem becomes fully known and accepted, as well as the implications of non-polynomial factorization, which is the new analysis technique needed for the proof, which shows the over 100 year error in number theory related to the use of the ring of algebraic integers.

But most importantly mathematics as a subset of logic becomes set along with my definition of mathematical proof, and the object ring takes its place as the one ring, helping to usher in an explosion of knowledge which may relate number theory to physics on a very deep level, and along with helping us understand our world better, help our species reach the stars.

EVERYTHING depended on me getting the truth through.

The future of the entire human species has been in the balance.

## Solving the factoring problem, easy

If you have a target composite T, and use

z^2 = y^2 + nT

where n is used to force z to be divisible by 3, then there will be an integer k such that

z = 3k/2.

But now find an integer x and odd prime p such that

x^2 = y^2 mod p

where also

2x = k.

So given k as defined before, you find an x that equals one half of it, and consider a prime p where

(k/2)^2 = y2 mod p.

Then it's trivial to show that

k^2 = 2^{-1}(nT) mod p

and, if f_1*f_2 = nT, then

f_1 = k mod p

and

f_2 = 2k mod p.

But that then shows that k, when a positive integer, must be greater than p, which is the crucial requirement. That is because

z = 3k/2

and

f_1 = k mod p

so if k is less than p, then f_1 = k, which contradicts with z having prime factors in common with k.

So how do I get the requirement on k?

Well given

x^2 = y^2 mod p and

2x = k, I multiply both sides of the latter by k and add to the first to get

x^2 + 2xk = k^2 + y^2 mod p

and then add k^2 to both sides to complete the square so that I have

(x+k)^2 = 2k^2 + y^2 mod p

and get back to my original equation with z = x+k, and nT = 2k^2 mod p.

Given that z = 3k/2, if I have integer factors f_1 and f_2, where

f_1*f_2 = nT

then

z = (f_1 + f_2)/2, so

k = (f_1 + f_2)/3.

The minimum positive integer k then is greater than or equal to 2sqrt(nT)/3 because that value is when f_1 = f_2.

Now then if you find a prime p less than the minimum positive value for k, where

2^{-1}(nT) mod p

is a quadratic residue modulo p, then the residue of k is found as

k^2 = 2^{-1}(nT) mod p

which follows from

nT = 2k^2 mod p.

So now you can search for k modulo p, but as you search up from the minimum possible k with the correct residue modulo p, you open up the minimum prime p, so, say, once you get to 10 times your original p, you find another prime p* that is 10 times the size of the previous one but still less than k for which

2^{-1}(nT) mod p*

is a quadratic residue, then you can now iterate upward from that point modulo that prime as well as the previous one. With only 10 primes used in this way, you would move up at least 10^100.

Which looks like a solution to the factoring problem.

Checking each k is easy enough, you just solve for z, with

z = 3k/2

and check to see if y is an integer, since y = sqrt(z^2 - nT).

That solution is trivially easy. There is not a real mathematician in the world who cannot trace out the steps of the proof.

The factoring problem is solved.

## JSH: It's a puzzle

I have to solve the problem and I'm running out of time. You all are part of a massive puzzle that I'm trying to unscramble and if I don't then nasty things happen, but more and more it looks like if I do, nasty things happen.

Part of the problem seems to be a natural conflict between evolution and civilization.

Civilization makes it easier for more people to survive, while evolution works best by killing off those who make stupid mistakes, so there is a continual battle of evolution against stability, without which humanity wouldn't exist as it would never have evolved.

Counter pressure against evolution seems to explain people like George W. Bush and Dick Cheney.

Their mistakes somehow don't matter. It has little impact on their survivability, but in a situation where mistakes don't matter, then negative memes can flourish until they reach a catastrophic tipping point, and stupidity once again causes death, and then evolution kicks in again with a vengeance.

Rising autism rates in the US may simply be the counter-force against evolution as humanity devolves slightly until stupidity reaches a level that civilization can no longer keep most people alive, which is where we are rapidly going.

So the answer may be that there is no solution but to wait, as inevitably as stupid people, like modern mathematicians keeping a one hundred plus year old error in place so that they can play at doing math, by controlling the world push, it to the brink, at which time necessarily there will be this huge die-off as evolution re-asserts itself.

Like how many of you really understand global warming?

My guess is that roughly 90% of you do NOT.

That probably is about a critical level above which humanity's devolution will be steep enough that it will no longer be able to maintain civilization at the level necessary to maintain stupid people like George W. Bush, or modern "pure math" mathematicians, where we're seeing critical pressure as hunger strikes much of the world.

But that can't be the full answer. And when is the critical point? It has to be within the next 10 years.

But is it possible for a 90% die-off rate? If so, then most of the people alive today will not be alive in 10 years, as evolution breaks the civilization barrier and once again forces forward development of the human race, and puts a negative on stupidity so that failure means death.
So evolution is the natural force antagonistic to civilization, and seeks to topple it.

And it is an irresistible force while civilization is a movable object.

The balance cannot be maintained as a species devolves when evolutionary pressures are removed until it loses the ability to maintain against those pressure whatever its solution.

Humanity has the solution of civilization so wrapped in the solution must be the key for evolution to re-assert, so we have a world now which is oddly tranquil in the face of global warming, well, because it's not "smart" enough to fully understand what's happening.

Which is evolution asserting itself to beat the solution of civilization.

Or maybe I need more brainstorming but I think I'm closing in on the answer, and it's not pretty.

## JSH: No simple answer then?

I posted as part of the initial brainstorming process about the oddity of high search rankings for topics related to my postings on my blogs versus low hit counts for those blogs as reported by Google Analytics and now Quantcast.

The issue here is not getting a lot of hits but an apparent contradiction, where, for instance, last time I checked in Google (just now): definition mathematical proof—brings up a page on my math blog at number five.

And in Google: define mathematical proof—brings up that page at number six.

I put those search strings that way to indicate that quotes are NOT to be used.

Google says there are 542,000 search results for that last search string, where my page comes up number six. But according to Google Analytics that page has received 33 hits in the past 30 days. And 3 hits today and 3 hits yesterday.

One early hypothesis I've come up with as I've begun brainstorming this problem is that there are two different data streams, as in there MUST be two different data streams to get such different results.

One data stream says you people lost as of course mathematicians have been ridiculing me and my research for years now with it clear that they wanted them dismissed. But the other data stream says you've won as research I've put out there for the public is not getting a lot of hit counts, as reported by two sources.

It gets weirder but I want go into how many different ways I look to measure influence on the world stage where I seem to have an impact (sometimes just to see, so your world may change just because I'm fiddling with something which is kind of humbling I guess) but then somehow, someway by other measure I have little to no impact at all!

One of my favorites was threatening lawsuits against Princeton and Harvard from angry parents who learn of fraud in math departments that was allowed by those universities so that endowments are nailed (I projected I think a 1/4 loss for both endowments) and coincidentally recently congressional investigations were started about, yup, HUGE, out-sized endowments at some universities, and as a good thing, some colleges decided to simply pay for their undergrads tuitions out of their endowments!!!

I LOVE coincidence and concurrent development, like when I came up with a business plan for YouTube and about two months later something similar (differences there were) was implemented.

If you consider the bottom-line of the impact of the reported hit results versus the search rankings there is only one major thing that is impacted: money.

Like there's a deliberate dampening, where it doesn't matter if I get high search rankings for things like defining mathematical proof, as long as I cannot make any money with my ideas.

Which gets us to DMESE where I gave away an idea for managed copy which would allow people to make copies, say, of high definition DVD's with an idea I GAVE AWAY, so why would I make money if it were implemented?

It, of course, has not been implemented and instead published reports indicate that the entertainment industry has given up on managed copy so YOU will not be able to legally copy your high definition DVD's, but that's a small price to mathematicians to pay for them to be able to keep teaching naive kids fake math.

Money would give me more leverage and more credibility while high search results leave the status quo.

Like the US in Iraq, the truth is not nearly so important any more, when too many people are willing to lose, and lose big, so that a few people can keep lying to them, and giving them nothing for something.

The rich have gotten richer in this country and screwed over everyone else. But people LOVE it!!!

They love starving, and dying and trying for nothing only to have at the end the bitter realization that they were simply conned by other people less scrupulous, and less believing in any kind of morality, who got rich.

You believe in them so they can teach your children junk, and oh yeah, keep you from legally copying your DVD's so Bush can get away with, oh, anything, including massive failure, and you can fight to put food into the mouths of your children, and gas into your car's tank.

You brought this on yourselves.

Hell on earth has been your own creation.

## JSH: No matter what, I win

Evolution gave me the best problem solving tools in the history of the human species.

And they are mine.

They will stay MINE.

I own them. And it's my math. Not yours. Not anyone else's.

It's MY MATH.

MINE.