## JSH: One more explanation, factoring solution

With posters yet again chortling victory against my research despite their failure to disprove anything about my solution to the factoring problem it's worth explaining again.

What I did was exploit a rational connection.

With rational solutions to

x^2 - Dy^2 = 1

I noticed you have rationals solutions to

(D-1)j^2 + (j+/-1)^2 = (x+y)^2

where j = ((x+Dy) -/+1)/D.

And that's it. 99% of the mathematics is right there in front of you, and covered with that initial statement. So how do I solve for x and y directly?

Well, I have TWO FACTORIZATIONS available:

(x-1)(x+1) = Dy^2

and

(D-1)j^2 = (x+y - (j+/-1))(x+y + (j+/-1))

So what I do is generally factor the second and I also found I needed to split up j, so I add variables: u and v

(x+y - (j+/-1)) = f_1*u

(x+y + (j+/-1)) = f_2*u*v^2

where f_1*f_2 = D-1.

And that's how v comes into the picture. Now recap: for EVERY rational solution to x^2 - Dy^2 = 1, you have a rational solution to:

(D-1)j^2 + (j+/-1)^2 = (x+y)^2

I note that if x = r/t and y = s/t, I have (r-t)(r+t) = Ds^2, and can consider a solution that factors D if it is an odd composite and g_1 and g_2 are non-trivial factors with:

r-t = g_1 and r+t = g_1, as then r= (g_1 + g_2)/2 and t = (g_1 - g_2)/2, and s = 1 or -1.

So rational solutions to r, s and t EXIST at a point that will factor D non-trivially.

One set of posters has repeatedly claimed they do not. With at least one claiming to have disproven that using the quadratic formula.

Now to guarantee non-trivial factorization of D, it suffices with non-zero r, s and t, for

abs(r-t) < D and abs(r+t) < D

and you'll notice I already showed at least one example of that case which must exist!

One set of posters have routinely claimed that both conditions cannot be simultaneously true.

Now I've noted that now you have a calculus problem of minimizing to find r, s and t as functions of v, such that you meet those conditions, and I've given ONE possible answer while to practically factor it may never be the case that you even need the s=1 or -1 case. But the proof of its existence shows that rational v is available in the desired range.

Now here are the explicit solutions for x and y:

y = [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

and

x = +/-(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v) - [+/-2Dv/(f_1 - f_2*v^2 - 2v) +/- 1 -/+(f_1 + f_2*v^2)/(f_1 - f_2*v^2 - 2v)]/(D-1)

where again f_1*f_2 = D-1, and the f's are non-zero integer factors, while v is a free variable.

To recap: what I did was exploit a connection between the factorization of D and the factorization of D-1.

Remarkably they are connected through Pell's Equation and an equation I derived from Pell's Equation using my Quadratic Diophantine Theorem (Google it).

Now for years I've claimed that math society has been ignoring major proofs of mine to hold on to the status quo as my research upsets HUGE swaths of established number theory, and now you have clear and irrefutable evidence in front of you of how far they will go in that denial.

Pell's Equation is one of the most famous in mathematical history.

The factoring problem is being used to supposedly secure the Internet.

With every security breach you read in the news, consider the possibility that factoring has been used, and that practitioners in various specialties are lying about it being broken, just like posters on these newsgroups lie about the efficacy of the equations above.

They do so to preserve their BELIEFS about the world in a way that makes them most comfortable without realizing the consequences of their belief system can be catastrophic. They are—religious about mathematics.

Factoring always had an easy answer: connect factoring one number to factoring another.

People just got it wrong for a while and now the truth is out. People make mistakes. That's not news. But please don't make the bigger mistake of continued denial to try and hold on to math ideas that just do not work.

Mathematics is a heartless discipline.

I know many of you have invested huge amounts of time and energy and years of your lives to learn mathematical ideas that if you're honest you'll have to realize are wrong.

But holding on to them will never make them right.