7の平方根
645751312...,\quad x_{5}={\frac {2081028097}{786554688}}=2.645751311064591...,\quad \dots }
All but the first of these satisfy the Pell's equation above.
The Babylonian method is equivalent to Newton's method for root finding applied to the polynomial x 2 − 7 {\displaystyle x^{2}-7} . The Newton's method update, x n + 1 = x n − f ( x n ) / f ′ ( x n ) , {\displaystyle x_{n+1}=x_{n}-f(x_{n})/f'(x_{n}),} is equal to ( x n + 7 / x n ) / 2 {\displaystyle (x_{n}+7/x_{n})/2} when f ( x ) = x 2 − 7 {\displaystyle f(x)=x^{2}-7} . The method therefore converges quadratically (number of accurate decimal digits proportional to the square of the number of Newton or Babylonian steps).