Efficient basic number-theoretic functions.

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MIT licensed by Daniel Fischer
Maintained by Carter Schonwald carter at wellposed dot com, Andrew Lelechenko andrew dot lelechenko at gmail dot com

Module documentation for

A library of basic functionality needed for number-theoretic calculations. The aim of this library is to provide efficient implementations of the functions. Primes and related things (totients, factorisation), powers (integer roots and tests, modular exponentiation).

This release supports GHC 8.0, 8.2, 8.4 and 8.6.

Breaking changes:

Remove 'Prime' type family and introduce 'Prime' newtype. This newtype
is now used extensively in public API:

primes :: Integral a => [Prime a]
primeList :: Integral a => PrimeSieve -> [Prime a]
sieveFrom :: Integer -> [Prime Integer]
nthPrime :: Integer -> Prime Integer

'sbcFunctionOnPrimePower' now accepts 'Prime Word' instead of 'Word'.

are no longer re-exported from 'Math.NumberTheory.Primes'.
Merge 'Math.NumberTheory.UniqueFactorisation' into
'Math.NumberTheory.Primes' (#135, #153).

From now on 'Math.NumberTheory.Primes.Factorisation.factorise'
and similar functions return [(Integer, Word)] instead of [(Integer, Int)].

Remove deprecated 'Math.NumberTheory.GCD' and 'Math.NumberTheory.GCD.LowLevel'.

Deprecate 'Math.NumberTheory.Recurrencies.*'.
Use 'Math.NumberTheory.Recurrences.*' instead (#146).

New features:

New functions 'nextPrime' and 'precPrime'. Implement an instance of 'Enum' for primes (#153):

> [nextPrime 101 .. precPrime 130]
[Prime 101,Prime 103,Prime 107,Prime 109,Prime 113,Prime 127]

Support Gaussian and Eisenstein integers in smooth numbers (#138).

Add the Hurwitz zeta function on non-negative integer arguments (#126).

Implement efficient tests of n-freeness: pointwise and in interval. See 'isNFree' and 'nFreesBlock' (#145).

Generate preimages of the totient and the sum-of-divisors functions (#142):

> inverseTotient 120 :: [Integer]

Generate coefficients of Faulhaber polynomials 'faulhaberPoly' (#70).


Better precision for exact values of Riemann zeta and Dirichlet beta
functions (#123).

Speed up certain cases of modular multiplication (#160).

Extend Chinese theorem to non-coprime moduli (#71).
This release supports GHC 7.10, 8.0, 8.2, 8.4 and 8.6.

Breaking changes:

Stop reporting units (1, -1, i, -i) as a part of factorisation
for integers and Gaussian integers (#101). Now `factorise (-2)`
is `[(2, 1)]` and not `[(-1, 1), (2, 1)]`.

Deprecate an old interface of 'Math.NumberTheory.Moduli.Sqrt'
and roll out a new one, more robust and type safe (#87).

Deprecate 'Math.NumberTheory.GCD' and 'Math.NumberTheory.GCD.LowLevel' (#80).
Use 'Math.NumberTheory.Euclidean' instead (#128).
Move 'splitIntoCoprimes' to 'Math.NumberTheory.Euclidean.Coprimes'.

Change types of 'splitIntoCoprimes', 'fromFactors' and 'prefFactors'
using newtype 'Coprimes' (#89).

Redesign API to modular square roots (#108)

Deprecate 'jacobi'' (#103).

Sort Gaussian primes by norm (#124).

Deprecate 'Math.NumberTheory.GaussianIntegers' in favor of

New features:

Implement Ramanujan tau function (#112):

> map ramanujan [1..10]

Implement partition function (#115):

> take 10 partition

Add the Dirichlet beta function on non-negative integer arguments (#120).
E. g.,

> take 5 $ Math.NumberTheory.Zeta.Dirichlet.betas 1e-15

Solve linear and quadratic congruences (#129).

Support Eisenstein integers (#121).

Implement discrete logarithm (#88).


Make return type of 'primes' and 'primeList' polymorphic instead of
being limited to 'Integer' only (#109).

Speed up factorisation of Gaussian integers (#116).

Speed up computation of primitive roots for prime powers (#127).
This release supports GHC 7.8, 7.10, 8.0, 8.2 and 8.4.

Breaking changes:

Remove 'Math.NumberTheory.Powers.Integer', deprecated in

Deprecate 'Math.NumberTheory.Primes.Heap'.
Use 'Math.NumberTheory.Primes.Sieve' instead.

Deprecate 'FactorSieve', 'TotientSieve', 'CarmichaelSieve' and
accompanying functions. Use new general approach for bulk evaluation
of arithmetic functions instead (#77).

Now 'moebius' returns not a number, but a value of 'Moebius' type (#90).

New functions:

A general framework for bulk evaluation of arithmetic functions (#77):

> runFunctionOverBlock carmichaelA 1 10

Implement a sublinear algorithm for Mertens function (#90):

> map (mertens . (10 ^)) [0..9]

Add basic support for cyclic groups and primitive roots (#86).

Implement an efficient modular exponentiation (#86).

Write routines for lazy generation of smooth numbers (#91).

> smoothOverInRange (fromJust (fromList [3,5,7])) 1000 2000


Now factorisation of large integers and Gaussian integers produces
factors as lazy as possible (#72, #76).
Switch to smallcheck 1.1.3.
This release supports GHC 7.8, 7.10, 8.0 and 8.2.

Breaking changes:

'Math.NumberTheory.Moduli' was split into

Functions 'jacobi' and 'jacobi'' return 'JacobiSymbol'
instead of 'Int'.

Functions 'invertMod', 'powerMod' and 'powerModInteger' were removed,
as well as their unchecked counterparts. Use new interface to
modular computations, provided by 'Math.NumberTheory.Moduli.Class'.

New functions:

Brand new 'Math.NumberTheory.Moduli.Class' (#56), providing
flexible and type safe modular arithmetic. Due to use of GMP built-ins
it is also significantly faster.

New function 'divisorsList', which is lazier than 'divisors' and
does not require 'Ord' constraint (#64). Thus, it can be used
for 'GaussianInteger'.


Speed up factorisation over elliptic curve up to 15x (#65).

Polymorphic 'fibonacci' and 'lucas' functions, which previously
were restricted to 'Integer' only (#63). This is especially useful
for modular computations, e. g., 'map fibonacci [1..10] :: [Mod 7]'.

Make 'totientSum' more robust and idiomatic (#58).
Switch to QuickCheck 2.10.
This release supports GHC 7.8, 7.10 and 8.0. GHC 7.6 is no longer supported.

Breaking changes:

Remove deprecated interface to arithmetic functions (divisors, tau,
sigma, totient, jordan, moebius, liouville, smallOmega, bigOmega,
carmichael, expMangoldt). New interface is exposed via
Math.NumberTheory.ArithmeticFunctions (#30).

Deprecate integerPower and integerWordPower from
Math.NumberTheory.Powers.Integer. Use (^) instead (#51).

Math.NumberTheory.Logarithms has been moved to the separate package
integer-logarithms (#51).

Rename Math.NumberTheory.Lucas to Math.NumberTheory.Recurrencies.Linear.

New functions:

Add basic combinatorial sequences: binomial coefficients, Stirling
numbers of both kinds, Eulerian numbers of both kinds, Bernoulli
numbers (#39). E. g.,

> take 10 $ Math.NumberTheory.Recurrencies.Bilinear.bernoulli
[1 % 1,(-1) % 2,1 % 6,0 % 1,(-1) % 30,0 % 1,1 % 42,0 % 1,(-1) % 30,0 % 1]

Add the Riemann zeta function on non-negative integer arguments (#44).
E. g.,

> take 5 $ Math.NumberTheory.Zeta.zetas 1e-15


Speed up isPrime twice; rework millerRabinV and isStrongFermatPP (#22, #25).
This release supports GHC 7.6, 7.8, 7.10 and 8.0.

Add Math.NumberTheory.ArithmeticFunctions with brand-new machinery
for arithmetic functions: divisors, tau, sigma, totient, jordan,
moebius, liouville, smallOmega, bigOmega, carmichael, expMangoldt (#30).
Old implementations (exposed via Math.NumberTheory.Primes.Factorisation
and Math.NumberTheory.Powers.Integer) are deprecated and will be removed
in the next major release.

Add Karatsuba sqrt algorithm, improving performance on large integers (#6).

Fix incorrect indexing of FactorSieve (#35).
This release supports GHC 7.6, 7.8, 7.10 and 8.0.

Add new cabal flag check-bounds, which replaces all unsafe array functions with safe ones.

Add basic functions on Gaussian integers.
Add Möbius mu-function.

Forbid non-positive moduli in Math.NumberTheory.Moduli.

Fix out-of-bounds error in Math.NumberTheory.Primes.Heap, Math.NumberTheory.Primes.Sieve and Math.NumberTheory.MoebiusInversion.
Fix 32-bit build.
Fix binaryGCD on negative numbers.
Fix highestPower (various issues).
Add integerLog10 variants at Bas van Dijk's request and expose
Math.NumberTheory.Powers.Integer, with an added integerWordPower.
Update for GHC-7.8, the type of some primops changed, they return Int# now
instead of Bool.
Fixed bugs in modular square roots and factorisation.
Relaxed dependencies on mtl and containers
Fixed warnings from GHC-7.5, Word(..) moved to GHC.Types
Removed SPECIALISE pragma from inline function (warning from 7.5, probably
pointless anyway)
Sped up factor sieves. They need more space now, but the speedup is worth it, IMO.
Raised spec-constr limit in MoebiusInversion.Int
Fixed Haddock bug
Added generalised Möbius inversion, to be continued
Added modular square roots and Chinese remainder theorem
Performance tweaks for powerModInteger (~10%) and
invertMod (~25%).
Fix bug in psieveFrom
Fix bug in nthPrime
Fix bug in powerMod
Relax bounds on array dependency for 7.4.*
Fix copy-pasto (only relevant for 7.3.*)
Fix imports for ghc >= 7.3
Added certificates and certified testing/factorisation
Fixed doc bugs
Elaborate on overflow, work more on native Ints in Eratosthenes
First release
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