number-theory

A number theory toolkit for JavaScript.

Functions

divisors(n)

Determines all of the divisors for a given number.

var divisors = require('number-theory').divisors;
divisors(6); // Returns [1, 2, 3, 6]

eulerPhi(n), totient(n)

Counts the positive integers less than a given number that are co-prime with the given number. For more information see the Wikipedia entry for Euler's Totient Function.

var phi = require('number-theory').eulerPhi;
phi(26); // Returns 12

factor(n)

Determines the prime factorization for a given integer. For more information see Wikipedia's Integer Factorization entry.

var factor = require('number-theory').factor;

/*
  Returns: [
    {  prime: 2, power: 2 },
    { prime: 3, power: 1 },
    { prime: 11, power: 1 }
  ]
*/
factor(132);

findDivisor(n)

Uses the Pollard-Rho integer factorization algorithm to quickly find a small divisor of the given number. Note: the divisor found need not be prime (as Pollar-Rho is a general integer factorization algorithm).

var findDivisor = require('number-theory').findDivisor;
findDivisor(152); // Returns 8

gcd(a, b)

Finds the greatest common divisor of two integers a and b.

var gcd = require('number-theory').gcd;
gcd(84, 172); // Returns 4

incMixed(tuple, bases)

Given a mixed-radix number and the bases for each digit, this determines the increment of the number. For more information, see Wikipedia's entry on Mixed Radix number systems.

var incMixed = require('number-theory').incMixed;

// A number representing a mixed-radix "clock" at 11:59 PM
var number = [59, 59, 23];

// The bases for each of the mixed radix digits (60 seconds to a minute,
// 60 minutes to an hour, 24 hours to a day).
var base = [60, 60, 24];

incMixed(number, base); // Returns [0, 0, 0] (or midnight the next day)

inverseMod(a, m)

Given an integer this function computes the modular multiplicative inverse to the given modulo.

var inverseMod = require('number-theory').inverseMod;
inverseMod(14, 17); // Returns 11

isAbundant(n)

Given an integer, returns a Boolean indicating whether it's an abundant number.

var isAbundant = require('number-theory').isAbundant;
isAbundant(36); // Returns true
isAbundant(35); // Returns false

isDeficient(n)

Given an integer, returns a Boolean indicating whether it's a deficient number.

var isDeficient = require('number-theory').isDeficient;
isDeficient(15); // Returns true
isDeficient(12); // Returns false

isHeptagonal(n)

Given an integer, returns a Boolean indicating whether it's a heptagonal number.

var isHeptagonal = require('number-theory').isHeptagonal;
isHeptagonal(112); // Returns true
isHeptagonal(175); // Returns false

isHexagonal(n)

Given an integer, returns a Boolean indicating whether it's a hexagonal number.

var isHexagonal = require('number-theory').isHexagonal;
isHexagonal(190); // Returns true
isHexagonal(50); // Returns false

isOctagonal(n)

Given an integer, returns a Boolean indicating whether it's an octagonal number.

var isOctagonal = require('number-theory').isOctagonal;
isOctagonal(65); // Returns true
isOctaongal(50); // Returns false

isPentagonal(n)

Given an integer, returns a Boolean indicating whether it's a pentagonal number.

var isPentagonal = require('number-theory').isPentagonal;
isPentagonal(92); // Returns true
isPentagona(50); // Returns false

isPerfect(n)

Given an integer, returns a Boolean indicating whether it's a perfect number.

var isPerfect = require('number-theory').isPerfect;
isPerfect(496); // Returns true
isPerfect(200); // Returns false

isPrime(n)

Determines if the given number is prime. Note: this is a particularly slow method that uses full prime factorization to determine if the number is prime. For a faster method see the miller function below.

var isPrime = require('number-theory').isPrime;
isPrime(7); // Returns true
isPrime(48); // Returns false

isSquare(n)

Given an integer, returns a Boolean indicating whether it's a square number.

var isSquare = require('number-theory').isSquare;
isSquare(16); // Returns true
isSquare(55); // Returns false

isTriangular(n)

Given an integer, returns a Boolean indicating whether it's a triangular number.

var isTriangular = require('number-theory').isTriangular;
isTriangular(21); // Returns true
isTriangular(25); // Returns false

jacobiSymbol(a, b)

Computes the Jacobi Symbol for the given numbers.

var jacobiSymbol = require('number-theory').jacobiSymbol;
jacobiSymbol(928, 33); // returns 1

logMod(a, b, m)

Solves a discrete logarithm. For more information see the following:

• Discrete Lograrithm
• Baby-step Giant-step algorithm

miller(n), isProbablyPrime(n)

Uses the determinisic Miller-Rabin Primality Test to determine if the given number is prime. Works for all positive integers less than 341,550,071,728,321.

var miller = require('number-theory').miller;
miller(17); // Returns true
miller(284); // Returns false

multiplyMod(a, b, m)

Multiplies the two given numbers mod the given modulus. See Wikipedia's entry on Modular Arithmetic.

var multiplyMod = require('number-theory').multiplyMod;
multiplyMod(928, 284, 18); // Returns 14

powerMod(base, exponent, mod)

Computes the power of a base mod the given modulus. For more information see Wikipedia's entry on Modular Exponentiation.

var powerMod = require('number-theory').powerMod;
powerMod(567283, 2843, 776); // Returns 299

primeFactors(n)

Computes a list of all prime factors for the given integer. Note: while this method fully computes the prime factorization of the integer, it only returns the primes and not the powers of the factorization. For full prime factorization please use factor.

var primeFactors = require('number-theory').primeFactors;
primeFactors(18); // Returns [2, 3]

primitiveRoot(m)

Computes the smallest primitive root for Z mod n, meaning a multiplicative generator for the group of units of Z mod n. For more information see Wikipedia's entry on Primitive roots modulo n.

var primitiveRoot = require('number-theory').primitiveRoot;
primitiveRoot(1043); // Returns 7

quadraticNonresidue(p)

Computes a quadratic nonresidue for the given number. For more information see Wikipedia's entry for Quadratic Residues.

var quadraticNonresidue = require('number-theory').quadraticNonresidue;
quadraticNonresidue(777); // Returns 5

randomPrimitiveRoot(m)

Find a random primitive root for Z mod n, meaning a multiplicative generator for the group of units of Z mod n. Unlike primitiveRoot, this function returns a random primitive root. For more information see Wikipedia's entry on Primitive roots modulo n.

sieve(n)

Determines a list of prime numbers up to the given bound by performing the Sieve of Eratosthenes.

var sieve = require('number-theory').sieve;
sieve(10); // Returns [ 2, 3, 5, 7 ]

squareRootMod(n, m)

Determines all square roots of a given number modulo the given modulus. For more information see Wikipedia's entry on Quadratic Residues.

var squareRootMod = require('number-theory').squareRootMod;
squareRootMod(1023, 77); // Returns [76, 1]

squareRootModPrime(n, p)

Uses the Tonelli–Shanks algorithm to determine a single square root in Z mod p.

var squareRootModPrime = require('number-theory').squareRootModPrime;
squareRootModPrime(100, 19) // Returns 9

Contributing

Pull requests are very welcome! If you see a function we're missing, have an alternate algorithm implementation, or even want to add a special case function we'd be delighted to review your code.

Try to stick to the following guidelines, as they will help get the PR merged and published quickly:

• New functions should be added to their own file under the lib/ directory
• Make sure to add an entry in the module.exports for new functions in the index.js file.
• Use two space characters per tab
• Please document your function using jsdoc (see any function in lib/ for an example on how to do this).
• Write a test for your function and place it in the tests/ folder with the same name that you gave for its lib/ counterpart.
• Add an entry to the documentation in this file (README.md). Also please try to keep the function list alphabetized for quick reference.

Thanks!

License

MIT