integer-roots

Calculating integer roots and testing perfect powers of arbitrary precision.

Integer square root

The integer square root (integerSquareRoot) of a non-negative integer \(n\) is the greatest integer \(m\) such that \(m\le\sqrt{n}\). Alternatively, in terms of the floor function, \(m = \lfloor\sqrt{n}\rfloor\).

For example,

> integerSquareRoot 99
9
> integerSquareRoot 101
10

It is tempting to implement integerSquareRoot via sqrt :: Double -> Double:

integerSquareRoot :: Integer -> Integer
integerSquareRoot = truncate . sqrt . fromInteger

However, this implementation is faulty:

> integerSquareRoot (3037000502^2)
3037000501
> integerSquareRoot (2^1024) == 2^1024
True

The problem here is that Double can represent only a limited subset of integers without precision loss. Once we encounter larger integers, we lose precision and obtain all kinds of wrong results.

This library features a polymorphic, efficient and robust routine integerSquareRoot :: Integral a => a -> a, which computes integer square roots by Karatsuba square root algorithm without intermediate Doubles.

Integer cube roots

The integer cube root (integerCubeRoot) of an integer \(n\) equals to \(\lfloor\sqrt[3]{n}\rfloor\).

Again, a naive approach is to implement integerCubeRoot via Double-typed computations:

integerCubeRoot :: Integer -> Integer
integerCubeRoot = truncate . (** (1/3)) . fromInteger

Here the precision loss is even worse than for integerSquareRoot:

> integerCubeRoot (4^3)
3
> integerCubeRoot (5^3)
4

That is why we provide a robust implementation of integerCubeRoot :: Integral a => a -> a, which computes roots by generalized Heron algorithm.

Higher powers

In spirit of integerSquareRoot and integerCubeRoot this library covers the general case as well, providing integerRoot :: (Integral a, Integral b) => b -> a -> a to compute integer \(k\)-th roots of arbitrary precision.

There is also highestPower routine, which tries hard to represent its input as a power with as large exponent as possible. This is a useful function in number theory, e. g., elliptic curve factorisation.

> map highestPower [2..10]
[(2,1),(3,1),(2,2),(5,1),(6,1),(7,1),(2,3),(3,2),(10,1)]