First off binary quadratic Diophantine equations are when you look for integer solutions to equations like:

c

_{1}x

^{2}+ c

_{2}xy + c

_{3}y

^{2}= c

_{4}+ c

_{5}x + c

_{6}y

Here x and y are the unknowns to be figured out. An example of such an equation is:

x

^{2}+ 2xy + 3y

^{2}= 4 + 5x + 6y

where I've simply used, c

_{1}= 1, c

_{2}= 2, c

_{3}= 3, c

_{4}= 4, c

_{5}= 5, c

_{6}= 6.

Such equations are also called binary quadratic forms, and in general can be reduced to a more basic binary quadratic form, like:

u

^{2}- Dv

^{2}= C

Here the letters used don't matter, of course, and in this case u and v represent the unknowns.

With my simple example above that reduction to the simpler form using my own research gives:

(-4(x+y) + 10)

^{2}+ 2s

^{2}= 166

So to fit into the form above:

u = -4(x+y) + 10 and v = s, while D = -2, and C = 166.

The reason to reduce to a simpler form is to aid in finding solutions. And in this case I can find the answers rather simply.

Subtracting 2s

^{2}from both sides the equation above is:

(-4(x+y) + 10)

^{2}= 166 - 2s

^{2}= 2(83 - s

^{2})

Running through possible odd s's I notice that s=9 works to give -4(x+y) + 10 = 2 or -2, so x+y = 2, or x+y = 3. And x = 4, y = -2, or x = 5, y = -2 work.

The letters do not matter so the general form can also be written as x

^{2}- Dy

^{2}= F.

That form has a modular solution for x and y.

Given x

^{2}- Dy

^{2}= F where all variables are non-zero integers:

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 derived here.

That result gives solutions to x

^{2}- Dy

^{2}= F mod N.

If modular arithmetic is unfamiliar to you, I have my own short introduction.

And that covers enough of the basic concepts of binary quadratic Diophantine equations to help in understanding my posts on the subject.

James Harris

## No comments:

Post a Comment