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As motivation, consider the square root of two. It is often approximated 1.414..., which some might incorrectly interpret as 1.41414141414..., or 140/99. Likewise, the reciprocal of the square root of two[?] to three decimal places[?] is 0.707, which is suggestive of 0.70707070..., or 70/99. If 70/99 approximates the reciprocal of the square root of two, it follows that 99/70 approximates the square root of two. As it turns out, the square root of two is between 140/99 and 99/70. The arithmetic mean of these two rationals is 19601/13860. That number squared is 384199201/192099600. It turns out that 2 times the denominator 192099600 is 384199200, which differs from the numerator by only one. p = 19601 and q = 13860 satisfies the Diophantine equation 2q^{2} + 1 = p^{2}. Any fraction of natural numbers p and q that satisfy this equation will be a reasonably good approximation for the square root of two.
More generally, if n is a given natural number, then any fraction of natural numbers p and q that satisfy Pell's equation
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