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Topic: Approximation to square roots
Replies: 7   Last Post: Jan 19, 2014 10:01 AM

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William Elliot

Posts: 2,637
Registered: 1/8/12
Re: Approximation to square roots
Posted: Jan 16, 2014 9:43 PM
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On Thu, 16 Jan 2014, wrote:

> "For each N,there exists a rational x / y such that the difference x / y -
> Sqr(N) is less than 1 / [2N^2.Sqr(N)]. x, y is a solution of Pell´s equation
> x^2 + N.y^2 = 1 ".

That's not Pell's equation and requirs any
such x,y to be complex or x = 1, y = 0.

Assume x^2 - n.y^2 = 1 has real roots, x,y.
Then a = -|x|, b = |y| is a solution to that Pell equation.
Thusly a/b - sqr n < 1/(2n^2 sqr n) as required.

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