^{2}- Dy

^{2}= F.

With a non-zero integer N for which a residue m exists where--m

^{2}= D mod N, and r, any residue modulo N for which Fr

^{-1}mod N exists then:

2x = r + Fr

^{-1}mod N and 2my = r - Fr

^{-1}mod N

It is easy to derive too. I use (x+my)(x-my) = F mod N, where m

^{2}= D mod N.

But the reduced form of what is called a binary quadratic Diophantine equation is supposedly well studied by mathematicians (I'm not a mathematician) but I haven't seen this simple solution. That's scary.

It's so easy I still puzzle myself about why it took me so long to notice it, but it has implications in some important areas where I hesitate to mention bigger ones, but for instance xy = T is a binary quadratic Diophantine equation.

That is, integer factorization is covered in this area, and can be considered to be simply solving a binary quadratic Diophantine equation. But the supposed difficulty of integer factorization for certain very large numbers is a reason for using it in web encryption. And, for instance, if you go to a site with "https" your computer sends something called a public key, which is worthless as a security system if someone could just quickly factor it.

I've now put a hold on research of mine which I worry might lead to a general way to solve binary quadratic Diophantine equations, and note that I'm not sure I could have achieved it. But mathematicians aren't acknowledging my results! It'd be irresponsible of me to continue further in such a situation.

Most people may not realize that mathematicians see their field as free from disruptions from sweeping changes. Unlike people in just about every other discipline out there they see mathematics as an ever growing

*mountain*of research, where their job is to build on what was done before--not overturn it.

To give an example the way I see it, as I got an undergraduate degree in physics, imagine if a group of people said that human transportation was by horse and buggy, and all we could do is build better and better horse drawn carriages? What if you drove a car by them and they just refused to acknowledge it existed?

And airplanes? Are you kidding? They'd probably laugh and tell you if vehicles ever fly, it will be if horses can first.

That to me is like this situation.

NO one else thinks like mathematicians.

For most people society is advancing, which means that some old ideas get overturned as better ways of doing things are found. But mathematicians pride themselves on the belief that no such thing happens in their field, and they refuse to acknowledge information that shows otherwise.

The breakdown of their worldview, however, in this area could have shattering repercussions for the world. While I am not saying my ideas definitely lead to a general way to solve binary quadratic Diophantine equations, there is enough done already for sensible people to have concerns.

But mathematicians are confident in their worldview and they don't approach these equations in this way!

If they turn out to be wrong, imagine, the news headlines blaring about a sudden collapse of the system used for encryption. Major companies around the world scrambling for some way to figure out how to continue doing business. Big tech giants find their continued existence questioned by pundits. And that's just on the corporate view without considering the country security issues. Or issues of personal security for most people.

And what would mathematicians say?

I suspect they'd be befuddled. Maybe even pathetic at that point. Insistent that these kinds of things just don't happen in their field.

And no one would care then.

I'm not interested in having anything to do with a new financial crisis that could take the world by surprise, but my not continuing research in this area doesn't mean someone else isn't.

But in my experience, I can assure you that mathematicians will calmly inform you that there is no point in worrying at all.

James Harris

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