arithmoi-0.2.0.5: Efficient basic number-theoretic functions. Primes, powers, integer logarithms.

Portability Non-portable (GHC extensions) Provisional Daniel Fischer Safe-Infered

Math.NumberTheory.Powers.General

Description

Calculating integer roots and determining perfect powers. The algorithms are moderately efficient.

Synopsis

# Documentation

integerRoot :: (Integral a, Integral b) => b -> a -> aSource

Calculate an integer root, `integerRoot k n` computes the (floor of) the `k`-th root of `n`, where `k` must be positive. `r = integerRoot k n` means `r^k <= n < (r+1)^k` if that is possible at all. It is impossible if `k` is even and `n < 0`, since then `r^k >= 0` for all `r`, then, and if `k <= 0`, `integerRoot` raises an error. For `k < 5`, a specialised version is called which should be more efficient than the general algorithm. However, it is not guaranteed that the rewrite rules for those fire, so if `k` is known in advance, it is safer to directly call the specialised versions.

exactRoot :: (Integral a, Integral b) => b -> a -> Maybe aSource

`exactRoot k n` returns `Nothing` if `n` is not a `k`-th power, `Just r` if `n == r^k`. If `k` is divisible by `4, 3` or `2`, a residue test is performed to avoid the expensive calculation if it can thus be determined that `n` is not a `k`-th power.

isKthPower :: (Integral a, Integral b) => b -> a -> BoolSource

`isKthPower k n` checks whether `n` is a `k`-th power.

isPerfectPower :: Integral a => a -> BoolSource

`isPerfectPower n` checks whether `n == r^k` for some `k > 1`.

highestPower :: Integral a => a -> (a, Int)Source

`highestPower n` produces the pair `(b,k)` with the largest exponent `k` such that `n == b^k`, except for `abs n <= 1`, in which case arbitrarily large exponents exist, and by an arbitrary decision `(n,3)` is returned.

First, by trial division with small primes, the range of possible exponents is reduced (if `p^e` exactly divides `n`, then `k` must be a divisor of `e`, if several small primes divide `n`, `k` must divide the greatest common divisor of their exponents, which mostly will be `1`, generally small; if none of the small primes divides `n`, the range of possible exponents is reduced since the base is necessarily large), if that has not yet determined the result, the remaining factor is examined by trying the divisors of the `gcd` of the prime exponents if some have been found, otherwise by trying prime exponents recursively.