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No. 1503:

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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 + 2b(n)2 b(n+1) = 2a(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).