[ { "name": "Arithmetic", "comment": " A library that provides useful number-theoretical functions for dealing\nwith integers, primes, divisibility, et cetera.\n\n# Primes\n@docs isPrime, primesBelow, primeFactors, primeExponents\n\n# Parity\n@docs isEven, isOdd\n\n# Divisors\n@docs divides, isMultipleOf, divisors, properDivisors, numDivisors\n\n# GCD and LCM\n@docs gcd, lcm, isCoprimeTo, totient\n\n# Base conversion\n@docs toBase, fromBase\n\n# Squares\n@docs intSquareRoot, exactIntSquareRoot, isSquare\n\n# Cubes\n@docs cbrt, intCubeRoot, exactIntCubeRoot, isCube\n\n# Modular arithmetic\n@docs powerMod\n\n", "unions": [], "aliases": [], "values": [ { "name": "cbrt", "comment": " Take the cube root of a number.\n\n cbrt 15.625 == 2.5\n", "type": "Float -> Float" }, { "name": "divides", "comment": " Test whether one number divides another.\n\n 10 `divides` 120 == True\n 10 `divides` 125 == False\n", "type": "Int -> Int -> Bool" }, { "name": "divisors", "comment": " Get all divisors of a number, in ascending order.\n\n divisors 20 == [1, 2, 4, 5, 10, 20]\n", "type": "Int -> List Int" }, { "name": "exactIntCubeRoot", "comment": " Integer cube root, returning `Nothing` if the given number is not a cube.\n\n exactIntCubeRoot 800 == Nothing\n exactIntCubeRoot 1000 == Just 10\n", "type": "Int -> Maybe Int" }, { "name": "exactIntSquareRoot", "comment": " Integer square root returning `Nothing` if the given number is not a\nsquare.\n\n exactIntSquareRoot 20 == Nothing\n exactIntSquareRoot 25 == Just 5\n", "type": "Int -> Maybe Int" }, { "name": "fromBase", "comment": " Interpret a list of digits as a number in the given base. The input is \nexpected to consist of integers `d` for which `0 <= d < base`.\n\n fromBase 2 [1, 0, 1, 0, 1, 0] = 42\n", "type": "Int -> List Int -> Int" }, { "name": "gcd", "comment": " Calculate the greatest common divisor of two integers. `gcd x 0` and\n`gcd 0 x` both return `x`. Negative arguments are made positive first.\n\n gcd 56 80 == 8\n", "type": "Int -> Int -> Int" }, { "name": "intCubeRoot", "comment": " Integer cube root, rounding down.\n\n intCubeRoot 800 == 9\n intCubeRoot 1000 == 10\n", "type": "Int -> Int" }, { "name": "intSquareRoot", "comment": " Integer square root, rounding down.\n\n intSquareRoot 20 == 4\n intSquareRoot 25 == 5\n", "type": "Int -> Int" }, { "name": "isCoprimeTo", "comment": " Test whether two integers are coprime.\n\n 56 `isCoprimeTo` 80 == False\n 5 `isCoprimeTo` 8 \n", "type": "Int -> Int -> Bool" }, { "name": "isCube", "comment": " Test whether a number is a cube.\n\n isCube 800 == False\n isCube 1000 == True\n", "type": "Int -> Bool" }, { "name": "isEven", "comment": " Test whether an integer is even.\n\n isEven 2 == True\n isEven 3 == False\n", "type": "Int -> Bool" }, { "name": "isMultipleOf", "comment": " Test whether one number is a multiple of another.\n\n 120 `isMultipleOf` 10 == True\n 125 `isMultipleOf` 10 == False\n", "type": "Int -> Int -> Bool" }, { "name": "isOdd", "comment": " Test whether an integer is odd.\n\n isOdd 2 == False\n isOdd 3 == True\n", "type": "Int -> Bool" }, { "name": "isPrime", "comment": " Test whether an integer is a positive prime.\n\n isPrime 2357 == True\n isPrime 500 == False\n", "type": "Int -> Bool" }, { "name": "isSquare", "comment": " Test whether a number is a square.\n\n isSquare 20 == False\n isSquare 25 == True\n", "type": "Int -> Bool" }, { "name": "lcm", "comment": " Calculate the least common multiple of two integers. `lcm x 0` and\n`lcm 0 x` both return `0`. Negative arguments are made positive first.\n\n lcm 56 80 == 560\n", "type": "Int -> Int -> Int" }, { "name": "numDivisors", "comment": " Get the number of divisors of a number (counting itself).\n", "type": "Int -> Int" }, { "name": "powerMod", "comment": " `powerMod b e m` efficiently calculates `b ^ e` (modulo `m`). It assumes\n`b >= 0`, `e >= 0` and `m >= 1`.\n\nFor example, to compute `4147 ^ 8671 % 1000` efficiently:\n\n powerMod 4147 8671 1000 == 803\n", "type": "Int -> Int -> Int -> Int" }, { "name": "primeExponents", "comment": " Return a list of all prime-exponent pairs for a given positive integer's\nprime decomposition, with the primes in ascending order. If the input is less\nthan 2, the empty list is returned.\n\n primeExponents 24 == [(2, 3), (5, 2)] -- 2^3 * 5^2\n primeExponents 531764 == [(2, 1), (11, 2), (13, 3)] -- 2^1 * 11^2 * 13^3\n primeExponents 1 == [] -- empty product\n", "type": "Int -> List ( Int, Int )" }, { "name": "primeFactors", "comment": " Return a list of all prime factors for a given positive integer, in\nascending order. If the input is less than 2, the empty list is returned.\n\n primeFactors 24 == [2, 2, 2, 3]\n primeFactors 767 == [13, 59]\n primeFactors 1 == []\n", "type": "Int -> List Int" }, { "name": "primesBelow", "comment": " Get all primes in the given range `[0..n-1]`, using the Sieve of\nEratosthenes.\n\n primesBelow 4 == [2, 3]\n primesBelow 17 == [2, 3, 5, 7, 11, 13]\n", "type": "Int -> List Int" }, { "name": "properDivisors", "comment": " Get all proper divisors (i.e., divisors less than the input) of a number,\nin ascending order.\n\n properDivisors 20 == [1, 2, 4, 5, 10]\n", "type": "Int -> List Int" }, { "name": "toBase", "comment": " Convert a number to a list of digits in the given base. The input number is\nmade positive first.\n\n toBase 2 42 = [1, 0, 1, 0, 1, 0] -- 42 in binary\n", "type": "Int -> Int -> List Int" }, { "name": "totient", "comment": " Compute Euler's totient function `φ(n)`: the number of positive integers\n`1 <= k <= n` for which `gcd(n, k) == 1`. The input is made positive first.\n\n totient 99 == 60\n totient 1450 == 560\n", "type": "Int -> Int" } ], "binops": [] } ]