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Sunday, May 08, 2011

Reducing binary quadratic Diophantines

As much as I talk about other key discoveries, I have talked less about my results about binary quadratic Diophantine equations, though a key one allows a one-step way to generally reduce binary quadratic Diophantine equations.

Given an equation of the form

c1x2 + c2xy + c3y2 = c4 + c5x + c6y

I've proven that you can reduce to

(A(x+y) - B)2 - As2 = B2 - AC

which is itself a binary quadratic with (x+y) and s unknowns, where

A = (c2 - 2c1)2 + 4c1(c2 - c1 - c3)

B = (c2 - 2c1)(c6 - c5) + 2c5(c2 - c1 - c3)

and

C = (c6 - c5)2 - 4c4(c2 - c1 - c3)

when neither A nor B equals 0.

As a first example let c1 = 1, c2 = 1, c3 = 1, c4 = 1, c5 = 1, c6 = 1, so:

x2 + xy + y2 = 1 + x + y

Which gives:

A = -3, B = -2, and C = 4

So the equation reduces to:

(-3(x+y) + 2)2 + 3s2 = 16

which works with s=0. So -3(x+y) + 2 = 4 or -4 should work, and the latter gives x+y=2, and you can substitute out x or y in the original equation to solve that way, where x=y=1 is a solution.

For a second example let c1 = 1, c2 = 2, c3 = 3, c4 = 4, c5 = 5, c6 = 6, so:

x2 + 2xy + 3y2 = 4 + 5x + 6y

Which gives:

A = -8, B = -20, and C = 33

And substituting and dividing 4 from both sides the equation reduces to:

(-4(x+y) + 10)2 + 2s2 = 166

which is:

(-4(x+y) + 10)2 = 166 - 2s2 = 2(83 - s2)

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 reducing form involves a completion of the square, as notice you can just multiply it out and also use the form:

A(x+y)2 - 2B(x+y) + C = s2

Note that form is necessary if A or B equals 0, because then you can't complete the square to get the other form.

This approach will work for any binary quadratic Diophantine equation except for if:

c2 - 2c1 = c2 - c1 - c3 = c6 - c5 = 0

However that special case is easily handled, for example:

x2 + 2xy + y2 = c4 + x + y

is: (x+y)2 - (x+y) - c4 = 0

And that form is equivalent to a unary case anyway, which is probably why my equations won't handle it. I worked out exactly why years ago, but don't remember enough to say that for certain.

I look at those equation every once in a while, just kind of wondering.

My method is to my knowledge the only known that turns reducing any binary quadratic Diophantine, except the special case as noted, into a one-step process.


James Harris
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