Power of simpler math

I found one of the best mathematical results in human history which is a simpler way to reduce things called binary quadratic Diophantine equations. That is, equations that look like this general case:

c₁x² + c₂xy + c₃y² = c₄ + c₅x + c₆y

where x and y are the unknowns to be figured out. A simple example of such an equation is:

x² + 2xy + 3y² = 4 + 5x + 6y

where I've simply used,  c₁ = 1, c₂ = 2, c₃ = 3, c₄ = 4, c₅ = 5, c₆ = 6.

With my simple example above I can reduce to a simpler form using my own research to get:

(-4(x+y) + 10)² + 2s² = 166

Now you can solve for x+y and s, and it's easier to find that s = 9 works, and x+y = 2 or 3, and x = 4, y = -2 is a solution. To see me work through in more detail, click here.

Now you may say, so what? Well, turns out we can immediately use this thing to get a simple result previously unknown before I discovered it, which is one of the greatest math finds of all time, and we're going to use it for some simple math trivia and approximate the square root of 2 with it.

And I start with a simple equation:

u² + Dv² = F.

And with my general method to reduce binary quadratic Diophantine equation we can find that:

(u-Dv)² + D(u+v)² = F(D+1)

And I'm now going to let D = -2, F = 1, since we're going after the square root of 2, and to make the equation look like a more familiar one I'm going to shift variables with: u=x, v=y, so my original is now:

x² - 2y² = 1

And now I can crank through my result to get:

(x+2y)² - 2(x+y)² = -1

And it's iterative! So I can do it again and again:

(3x+4y)² - 2(2x + 3y)² = 1

Next is:

(7x + 10y)² - 2(5x + 7y)² = -1

And iterating one more time:

 (17x + 24y)² - 2(12x + 17y)² = 1

And the more astute of you may have noticed that x = 1, y = 0 is a solution to the original equation, so guess what? We've solved the original equation as well with JUST my research result. Using that on the last:

(17)² - 2(12)² = 1

And we can still iterate, but let's do it now with just the numbers.

So now for another iteration: x = 17, y = 12,

so: (3(17)+4(12))² - 2(2(17) + 3(12))² = -1

Which is: (99)² - 2(70)² = -1

And that gives the slightly more impressive approximation of 99/70 is about: 1.4142

And if you're bored you can just keep going! Where now x=99 and y=70. And it works out to infinity with ever more precise approximations to sqrt(2).

Next one is: x=577, y=408, and 577/408 is approximately 1.41421.

Why do these solutions approximate the positive square root of 2?

Because x² - 2y² = 1, is:

(x² - 1)/y² = 2, and you can just take the square root of both sides now:

sqrt(x² - 1)/y = sqrt(2), so the trick then is that approximately x/y = sqrt(2).

And x² - 2y² = 1 was used over a thousand years ago, and one of its uses was, yup, approximating square roots, and I wonder if I'd have been cheered if I showed some of the ancients my simple result above?

I've used my result with much bigger things though.


James Harris