# integer-roots

Integer roots and perfect powers

https://github.com/Bodigrim/integer-roots

 Version on this page: 1.0.0.1 LTS Haskell 21.22: 1.0.2.0@rev:1 Stackage Nightly 2023-12-01: 1.0.2.0@rev:1 Latest on Hackage: 1.0.2.0@rev:1

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#### Module documentation for 1.0.0.1

• Math
• Math.NumberTheory
Depends on 2 packages(full list with versions):
Used by 2 packages in lts-18.8(full list with versions):

# 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 . Alternatively, in terms of the floor function, .

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 .

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)]

# 1.0.0.1

• Compatibility fixes for GHC 9.2.

# 1.0

• Initial release.