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Recited from memory by: unstattedCommoner

**The author writes:**

The approximation sqrt(2) ~= 665857/470832 was obtained by carrying out three iterations of the recurrence relation

*a*(*n*+1) = *a*(*n*)^{2} + 2*b*(*n*)^{2}
*b*(*n*+1) = 2*a*(*n*)*b*(*n*)

starting with *a*(0) = 3 and *b*(0) = 2. *x*(*n*) = *a*(*n*)/*b*(*n*) is then the approximation to sqrt(2).

Convergence of this method is extremely rapid:

*x*(*n*) - sqrt(2) is roughly proportional to ((*x*(0)-sqrt(2))/(2 sqrt(2)))^{(2n)}.