Copyright | (c) 2011 Daniel Fischer |
---|---|

License | MIT |

Maintainer | Daniel Fischer <daniel.is.fischer@googlemail.com> |

Stability | Provisional |

Portability | Non-portable (GHC extensions) |

Safe Haskell | None |

Language | Haskell98 |

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

- integerRoot :: (Integral a, Integral b) => b -> a -> a
- exactRoot :: (Integral a, Integral b) => b -> a -> Maybe a
- isKthPower :: (Integral a, Integral b) => b -> a -> Bool
- isPerfectPower :: Integral a => a -> Bool
- highestPower :: Integral a => a -> (a, Int)
- largePFPower :: Integer -> Integer -> (Integer, Int)

# Documentation

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

Calculate an integer root,

computes the (floor of) the `integerRoot`

k n`k`

-th
root of `n`

, where `k`

must be positive.
`r = `

means `integerRoot`

k n`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`

,

raises an error. For `integerRoot`

`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.

isKthPower :: (Integral a, Integral b) => b -> a -> Bool Source

checks whether `isKthPower`

k n`n`

is a `k`

-th power.

isPerfectPower :: Integral a => a -> Bool Source

checks whether `isPerfectPower`

n`n == r^k`

for some `k > 1`

.

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

produces the pair `highestPower`

n`(b,k)`

with the largest
exponent `k`

such that `n == b^k`

, except for

,
in which case arbitrarily large exponents exist, and by an
arbitrary decision `abs`

n <= 1`(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.

largePFPower :: Integer -> Integer -> (Integer, Int) Source

produces the pair `largePFPower`

bd n`(b,k)`

with the largest
exponent `k`

such that `n == b^k`

, where `bd > 1`

(it is expected
that `bd`

is much larger, at least `1000`

or so), `n > bd^2`

and `n`

has no prime factors `p <= bd`

, skipping the trial division phase
of

when that is a priori known to be superfluous.
It is only present to avoid duplication of work in factorisation
and primality testing, it is not expected to be generally useful.
The assumptions are not checked, if they are not satisfied, wrong
results and wasted work may be the consequence.`highestPower`