JSH: Good news!

Rather than just talk about my latest surrogate factoring idea and worry about consequences I've been playing with it, using small primes, and it works.

x^2 = y^2 mod T

and

k^2 = 2xk mod T

define a simple family of relations that allow you to factor a target composite T, by instead factoring some other number that I call the surrogate. The smart ones among you know just by looking that with all integers, for any integer k there must exist x and y mod T.

That means this remarkably simple idea has to work mod T, so once you pick a k, all you can do is move things around in multiples of T, as you look for non-trivial factorizations.

For the more brilliant among you, the surest test of this idea is asking, how could I find solutions for x and y using k, if I already knew the factorization of T?

That's one of the things I puzzled over for months as I considered this idea from a distance, and mostly worked on theory.

Oh yeah, for those of you who wondered, no, there is no way that mathematicians of the world could NOT have some clue that a lot of what they were doing was bogus when they are clearly trying their very best to sit quietly on this information, hoping nothing will happen.

As, hey!!! A person can't invent a new way to factor and it not get noticed.

Even back in the 1800's this would have been buzzing around the world.

So now there is no doubt—supposedly top mathematicians around the world MUST have known on some level that what they were doing was bogus.

Like, imagine if say, Andrew Wiles were serious about desperately wanting a real answer for Fermat's Last Theorem and really believed he'd found one, if there were a hint of a clue that he was wrong, would he hide from it? Or would he chase it down to be sure?

What if instead he and his colleagues just tried to deny, until brought down by brilliant research in another area, where they still at first tried to deny?

Could anyone doubt that they were cons all along?

It is EASY to lie about mathematics. It is a magnet for cons.

The real math is done by the people who work in practical areas, doing things like building better cars, faster computer chips, and better plastics. The "pure math" people are mostly not doing anything at all, and at this late stage in the game, they are completely controlled by con artists who know how to play at being real mathematicians though they do no important research at all…

And so many of you were so easily conned, by people who now have to sit still, hoping against hope, that the world ignores brilliant new knowledge to its own detriment, or they are found out...

A reply to someone who wrote that the values of X and Y in the examples do not satisfy the congruence, and so the presence of factors of T in X+Y or X-Y are due to something else.

Read the theory carefully. Mathematically you can do most of that theory with a prime factor p of T, and the smaller the k, that is, the lower the k/T ratio, the more you squeeze out the larger factors.

x^2 = y^2 mod p

and

k^2 = 2xk mod p

will do just fine to non-trivially factor.

I use small k's because it's easier for me to factor them.

With surrogates that are fully factored and the small numbers currently being used, it is factoring somewhere between 50% and 33% of the time, which is of course not random.

It is not random. If you test it properly then it is clearly not random.

People claiming otherwise either are screw-ups or liars.

I'm pushing people more towards my math blog and Extreme Mathematics group now, where I have started putting up examples.

The best overview on the planet of surrogate factoring is probably at my group:

Kind of wild, I guess, and it is cool having invented my own factoring method.

It may be key to ending the Math Wars.

And finally, the world will finally know that they were fought, how they were fought, and why…

Surrogate factoring, some speculation

I am kind of in shock as I play with small numbers still and find that my latest approach to what I call surrogate factoring follows theory and works remarkably well.

And remember I just extended from congruence of squares just slightly so that I have two primary relations:

x^2 = y^2 mod T

and

k^2 = 2xk mod T

where T is the target composite, and k is the control variable, which you use to get the surrogate. You factor the surrogate and use its factors to find x and y—creatively solving the difference of squares.

Works great with the small numbers I'm testing.

Now I have quite a few mathematical results by now, and even did some clean-up work in logic and gave the definition of mathematical proof.

It isn't that big a surprise for me to be able to find a simple approach to factoring, and come up with a new factoring method. I found a short proof of Fermat's Last Theorem, years ago. It is not a surprise that I can push the envelope on factoring, and unfortunately, it's not a surprise for mathematicians around the world to try and ignore it.

Problem is "pure math". Turns out that in history you get these freaks of nature who figure out mathematics and there are only a few of them, but mathematics is very important, so as it has gained in prestige LOTS more people want in on the status than can possibly figure out important mathematical results.

More people want in on the prestige than can figure out important mathematics that justifies the prestige.

And going to school for decades cannot make you able to figure out a simple way to extend factoring. Who knows what makes a person that can.

So what to do if you want the prestige anyway?

Well, easy answer, do useless work that has no real mathematical value, but just make sure it is not related to anything important—it is "pure"—and as long as everyone with you is in on the game, it will work out just fine.

And today lots more people are supported by mathematics than could possibly be supported if they were forced to do real research.

My guess is that by now over 99% of the mathematical research being done in "pure math" areas is completely wrong.

So what does that have to do with factoring?

Well people who learn to do useless math can come up with what they think are brilliant ideas that maybe are not, like a clever way that they hope can secure the world's information systems, because to them, factoring is a hard problem, but what if it isn't?

So here we are. Every time I talk a bit about surrogate factoring coincidentally there is a correction on world stock markets of some sort, which I say is coincidental because it seems too far fetched even to me that many people understand yet what is happening, but the recent stock market troubles gives you some clue about how far these mathematicians are willing to go.

I think they'd let the world stock markets crash and burn, and let civilization itself crumble before they'd admit the truth about what they are doing.

I think they'd let every OTHER human being on the planet die a miserable death before they'd tell the truth.

I think they'd sacrifice every one else but themselves if they thought that'd allow them to lie just one more day.

But that is speculation on my part.

Discussion on surrogate-factoring

Theory is one thing, but examples can help gather interest after mathematical proof, as I guess there is just something about seeing an actual result that makes a difference even when something has been proven, so here are some examples with surrogate factoring using some small numbers.

With T = 599293, I picked k with a k/T ratio of 1/30, which means that k is approximately T/30, and found a factorization with k=19978, where I got

y=207419/2, and x=175011/2

so x+y=191215, which has 229 as a factor, and T = (229)(2617).

That took 7 iterations from the starting k.

My next example shows a rather bad run, where with T = 1972897, k = -65765, I have

y=387935/2, and x=561737/2

so x-y=86901, which has 349 as a factor and T = (349)(5653).

And that took a whopping 114 iterations from my starting k, but I'm just picking some primes multiplying them out and seeing what I get so I give that rather bad result.

Finally I have T = 2066147, where k=68873, works giving me

y=-430137/2, and x=334429/2

so, x+y=-47854, which has 337 as a factor and T = (337)(6131).

And it took 9 iterations.

For my examples I took a table of primes and just picked one 3 digit prime and one 4 digit prime, multiplied them together and factored with my surrogate factoring method, where I picked the starting k using

k = floor(T/30)

with a-b=-1, and incremented by 1, until something worked. For two cases that worked rather well, while for one it took a lot, and I didn't throw out any results but just sat here and did 3 in a row.

If you read my earlier posts you may know that a k/T ratio of 1/30 should not be optimal anyway, but smaller k's are easier to factor, and I think you can see that even there, the odds are not that terrible.

So I extended mathematics in the area of factoring with a simple idea where for years I've wondered why you couldn't factor a target composite by instead factoring a surrogate. In each of the examples above the surrogate that is being factored is

2k^2 - T

and by factoring that number I get x and y as explained in prior postings and in that way factor the target composite.

Notice then that surrogate factoring is another way to get a difference of squares.

It is a highly creative way to do so, using some remarkably simple algebra and very basic theory as explained in prior post, so why wasn't it thought of before?

Who knows? I just thought a couple of years ago that maybe there was some way to factor a target by instead factoring some other number, and wouldn't that be neat!

It is a simple idea where most of the mathematics is simple and it may be possible to turn this into an effective practical approach, so mathematicians and cryptographers around the world should be buzzing about this idea, I'd think.

But then again, I have lots of mathematical discoveries that should create buzz, and have had them for years.

Instead of cheering my research, mathematicians call me a crackpot.

Go figure.

But, then again, they are proud those people, and certain of their brilliance.

Maybe I just make them feel stupid, so they choose to ignore simple answers, no matter where I find them.

Surrogate factoring and the k/T ratio

For years I've done research on ways you might factor a target composite T by factoring some other number I call the surrogate, and after a lot of failed approaches I realized that the idea mathematically reduced to a couple of very simple relations:

x^2 =E2=89=A1 y^2 mod T

and

k^2 =E2=89=A1 2xk mod T

where the first should be familiar enough, while the second is an addition needed mathematically by the concept of surrogate factoring.

So after a lot of years of fumbling around I found mathematically I could reduce the idea quite simply to the given relations.

Now using those requires going to explicit equations:

x^2 =3D y^2 + aT

and

k^2 =3D 2xk + bT

and I can add one to the other, and complete the square to find:

(x+k)^2 =3D y^2 + 2k^2 + (a-b)T

So, if you have

4f_1f_2 =3D 2k^2 + (a-b)T

then y =3D f_1 - f_2 and

x =3D f_1 + f_2 - k

and now the idea is just slightly more complicated, and you can see that actually trying to factor with it requires that you pick a-b and k=2E

Some analysis I won't go into here indicates that the value of a-b should be 1 or -1, so things got simpler there, but now that just leaves k.

So after all that research—the product of years of effort on my part distilling this idea down—I have one variable—k—which controls the outcome.

I have talked about this idea and approach on the newsgroups before as I came across it last year, with a few problems still figuring out details, but the essence has been around for months, and there were posters claiming it worked no better than random.

I refused to test it myself while I considered theoretical issues and worried about other things, but finally sat down and checked myself, and noticed they were wrong, but there is some interesting behavior around how big k is relative to T—the k/T ratio.

That is, I checked with very small primes—to see if the idea worked at all—so I used three digit primes, and found that it factored remarkably well, but I did do other checks and saw it factoring rather badly, though it still would factor.

Thinking about it, I realized that the k/T ratio was dropping, and that when it was factoring very well the k/T ratio was about 0.2 or 20%. But I don't know if that is the prime value, as the research is very rough at this point. But an optimal k/T ratio seems key.

The lower the ratio the better as the surrogate you factor is 2k^2 + (a-b)T, so the smaller it is, the better, so I used a-b=3D-1, and a k that was about 20% of T, and got great results.

I mean, like it factored as I expected, and clearly was not random, so those other posters either screwed up or lied. But that was with the product of three digit primes.

It is a new factoring technique that follows from that simple concept, and extends from the basic congruence of squares with only a single other congruence, from which follows some rather basic equations where a little analysis indicates you need only change one variable—k—and that the ratio of that variable to your target composite T, may be the key to this idea.

For more information on the earlier analysis (I don't have anything on my current research like the k/T ratio) go to my Extreme Mathematics group: 