Integer square root

In number theory, the integer square root (isqrt) of a positive integer n is the positive integer m which is the greatest integer less than or equal to the square root of n,

$\mbox{isqrt}( n ) = \lfloor \sqrt n \rfloor.$

For example, $\mbox{isqrt}(27) = 5$ because $5\cdot 5=25 \le 27$ and $6\cdot 6=36 > 27$.

Algorithm

One way of calculating $\sqrt{n}$ and $\mbox{isqrt}( n )$ is to use Newton's method to find a solution for the equation $x^{2} - n = 0$, giving the iterative formula

${x}_{k+1} = \frac{1}{2}\left(x_k + \frac{ n }{x_k}\right), \quad k \ge 0, \quad x_0 > 0.$

The sequence $\{ x_k \}$ converges quadratically to $\sqrt{n}$ as $k\to \infty$. It can be proven that if $x_{0} = n$ is chosen as the initial guess, one can stop as soon as

$| x_{k+1}-x_{k}| < 1$

to ensure that $\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor.$

Using only integer division

For computing $\lfloor \sqrt n \rfloor$ for very large integers n, one can use the quotient of Euclidean division for both of the division operations. This has the advantage of only using integers for each intermediate value, thus making the use of floating point representations of large numbers unnecessary. It is equivalent to using the iterative formula

${x}_{k+1} = \left\lfloor \frac{1}{2}\left(x_k + \left\lfloor \frac{ n }{x_k} \right\rfloor \right) \right\rfloor, \quad k \ge 0, \quad x_0 > 0, \quad x_0 \in \mathbb{Z}.$

By using the fact that

$\left\lfloor \frac{1}{2}\left(x_k + \left\lfloor \frac{ n }{x_k} \right\rfloor \right) \right\rfloor = \left\lfloor \frac{1}{2}\left(x_k + \frac{ n }{x_k} \right) \right\rfloor,$

one can show that this will reach $\lfloor \sqrt n \rfloor$ within a finite number of iterations.

However, $\lfloor \sqrt n \rfloor$ is not necessarily a fixed point of the above iterative formula. Indeed, it can be shown that $\lfloor \sqrt n \rfloor$ is a fixed point if and only if $n + 1$ is not a perfect square. If $n + 1$ is a perfect square, the sequence ends up in a period-two cycle between $\lfloor \sqrt n \rfloor$ and $\lfloor \sqrt n \rfloor + 1$ instead of converging.

Domain of computation

Although $\sqrt{n}$ is irrational for many $n$, the sequence $\{ x_k \}$ contains only rational terms when $x_0$ is rational. Thus, with this method it is unnecessary to exit the field of rational numbers in order to calculate $\mbox{isqrt}( n )$, a fact which has some theoretical advantages.

Stopping criterion

One can prove that $c=1$ is the largest possible number for which the stopping criterion

$|x_{k+1} - x_{k}| < c\$

ensures $\lfloor x_{k+1} \rfloor=\lfloor \sqrt n \rfloor$ in the algorithm above.

In implementations which use number formats that cannot represent all rational numbers exactly (for example, floating point), a stopping constant less than one should be used to protect against roundoff errors.