3. Reducing binary quadratic Diophantine equations
Turns out that equations of the form:
c1x2 + c2xy + c3y2 = c4 + c5x + c6y
where the c's are known and x and y are unknown are called binary quadratic equations, and Diophantine means you're looking for integer solutions. An example of such an equation is:
x2 + 2xy + 3y2 = 4 + 5x + 6y
It has solutions x = 4, y = -2, or x = 5, y = -2, which I found from a simpler form:
(-4(x+y) + 10)2 + 2s2 = 166
Found using my own research.
Few people can claim to have improved on Gauss but my method is more straightforward than prior techniques where you need to use a discriminant. Simpler, easier--it is then the best in the world:
That seems good enough for it come in at #3 on my personal scale of most compelling results.
2. Modular solution to binary quadratic Diophantine equations
For me one of my most surprising results because of its simplicity came from just noticing something rather obvious with equations of the form: x2 - Dy2 = F
I realized that in modular arithmetic you could always factor and in so doing solve for x and y modulo some N:
x2 - Dy2 = F = (x - my)(x + my) mod N
Where: m2 = D mod N
Given any nonzero integer D, there exists an N for which it is a quadratic residue.
Find 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
One of my easiest results to derive it's also one that could have been known in the time of Gauss, so why am I the guy talking about it in the 21st century?
That's just weird. I keep wondering about this one, highly suspicious that it's already out there.
But then I can do some things with it, like in this blog post also a paper:
For those reasons it easily comes in at #2.
1. Non-polynomial factorization
Who knew that breaking from the mold with factoring polynomials would turn into a wild adventure. This result of mine is easily the most controversial as well as most tested, well-worked with the wackiest wildest story. Everything about it, including that the very first post on this blog is related to it, make it by far the #1 most compelling result.
You see, I got bored with polynomial factorization and figured out a way to factor a polynomial into non-polynomials:
7(175x2 - 15x + 2) = (5a1(x) + 7)(5a2(x)+ 7)
where the a's are roots of
a2 - (7x-1)a + (49x2 - 14x) = 0
The techniques used to derive my method for reducing binary quadratic Diophantine equations were developed deriving that non-polynomial factorization. Reality is, I played with binary quadratic Diophantine equations to build confidence in those methods. Things felt that weird. I had to find something else to build confidence. Those techniques are my most ridiculed and criticized of all. And represent possibly the area where people come out of the woodwork with the greatest effort to discredit anything I have. So there is no doubt about that position at a solid #1.
And those are my personal top three. Others might have a different set, but for me, these are the ones.