number-theory-2

A lightweight and modern number theory library for JavaScript and Node.js.
It includes common number theory utilities all written with clarity and BigInt support.

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Installation

npm install number-theory-2

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Features

• Works with both Number and BigInt
• Pure JavaScript (no dependencies)
• Simple, modular, and easy to extend
• Accurate for very large integer computations

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Usage

import { gcd, mod, fibonacci } from "number-theory-2";

console.log(gcd(84, 18)); // ➜ 6n
console.log(mod(-10, 3)); // ➜ 2n
console.log(fibonacci(10)); // ➜ [0n, 1n, 1n, 2n, 3n, 5n, 8n, 13n, 21n, 34n, 55n]

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Currently Available Functions

Arithmetic

• ceilDiv(a, b) – Integer ceiling division ceil(a / b) with correct signed behavior
• crt(remainders, moduli) – Chinese Remainder Theorem solution for coprime moduli
• discreteLog(a, b, m) – Solves the discrete logarithm problem aᵡ ≡ b (mod m)
• divides(a, b) – Checks if a divides b (throws if a = 0)
• divisors(n) – Returns all positive divisors of a given integer n
• extendedGCD(a, b) – Extended Greatest Common Divisor
• floorDiv(a, b) – Euclidean division returning { q, r } with 0 ≤ r < |b|
• fromContinuedFraction(arr) – Returns rational { n, d } from continued fraction coefficients
• gcd(a, b) – Greatest Common Divisor using Euclid’s algorithm
• gcdArray(arr) – Computes the GCD of an array of numbers
• isEven(n) – Checks if a number n is even
• isOdd(n) – Checks if a number n is odd
• isPrimitiveRoot(g, n) – Checks if g is a primitive root modulo n
• lcm(a, b) – Least Common Multiple using the formula lcm(a, b) = |a*b| / gcd(a, b)
• lcmArray(arr) – Computes the LCM of an array of numbers
• leastAbsoluteResidue(a, m) – Returns minimal absolute residue in (-m/2, m/2] (tie at m/2 resolves to +m/2)
• mod(a, b) – Modular remainder (handles negative numbers and BigInts)
• modAdd(a, b, m) – Modular addition (a + b) mod m with normalization
• modDiv(a, b, m) – Modular division a * b⁻¹ (mod m); requires gcd(b, m) = 1
• modInverse(a, m) – Modular multiplicative inverse of a under modulo m
• modInverseArray(arr, m) – Efficiently computes the modular inverses of an array of numbers
• modMul(a, b, m) – Modular multiplication (a * b) mod m with normalization
• modSqrt(a, p) – Modular square root x such that x² ≡ a (mod p)
• modSub(a, b, m) – Modular subtraction (a - b) mod m with normalization
• orderMod(a, m) – Multiplicative order: smallest k such that a^k ≡ 1 (mod m)
• powMod(base, exp, mod) – Fast modular exponentiation (base^exp) mod mod
• primitiveRoot(n) – Finds the smallest primitive root modulo n
• sign(n) – Returns the sign of a number n
• solveCongruence(a, b, m) – Solves a·x ≡ b (mod m) canonically
• toContinuedFraction(n, d) – Returns continued fraction coefficients for n/d

Combinatorics

• bell(n) – Computes the n-th Bell number
• catalan(n) – Computes the n-th Catalan number
• combination(n, r) – Computes the number of combinations of n items taken r at a time
• doubleFactorial(n) – Computes the double factorial n!!
• factorial(n) – Computes the factorial of n (n!)
• fibonacci(n) – Generates Fibonacci sequence up to the n-th term
• hexagonal(n) – Computes the n-th hexagonal number
• isHexagonal(n) – Checks if a number n is hexagonal
• isPentagonal(n) – Checks if a number n is pentagonal
• isPolygonal(s, n) – Checks if a number n is s-gonal
• isSquare(n) – Checks if a number n is square
• isTriangular(n) – Checks if a number n is triangular
• lucas(n) – Generates Lucas sequence up to the n-th term
• motzkin(n) – Generates Motzkin sequence up to the n-th term
• nthFibonacci(n) – Returns the n-th Fibonacci number
• nthLucas(n) – Returns the n-th Lucas number
• nthMotzkin(n) – Returns the n-th Motzkin number
• nthPadovan(n) – Returns the n-th Padovan number
• nthPerrin(n) – Returns the n-th Perrin number
• nthTribonacci(n) – Returns the n-th Tribonacci number
• padovan(n) – Generates Padovan sequence up to the n-th term
• pascalRow(n) – Returns the n-th row of Pascal's triangle
• pentagonal(n) – Computes the n-th pentagonal number
• perrin(n) – Generates Perrin sequence up to the n-th term
• permutation(n, r) – Computes the number of permutations of n items taken r at a time
• polygonal(s, n) – Computes the n-th s-gonal number
• square(n) – Computes the n-th square number
• subfactorial(n) – Computes derangements !n (subfactorial)
• triangular(n) – Computes the n-th triangular number
• tribonacci(n) – Generates Tribonacci sequence up to the n-th term

Function

• aliquotSum(n) – Computes the Aliquot Sum of a number n
• carmichael(n) – Computes the Carmichael function λ(n)
• collatzSequence(n) – Generates the Collatz sequence from n down to 1
• collatzSteps(n) – Counts steps to reach 1 under the Collatz process
• cototient(n) – Computes cototient n − φ(n)
• dedekindPsi(n) – Computes the Dedekind psi function ψ(n)
• digitalRoot(n) – Computes the digital root of an integer n
• isAbundant(n) – Checks if a number n is Abundant
• isDeficient(n) – Checks if a number n is Deficient
• isPerfect(n) – Checks if a number n is Perfect
• isSquareFree(n) – Checks if a number n is square-free
• jordanTotient(n, k) – Computes Jordan's totient function J_k(n)
• liouville(n) – Computes the Liouville function of a number n
• mobius(n) – Computes the Möbius function μ(n)
• omegaBig(n) – Computes the number of prime factors of n (big omega)
• omegaSmall(n) – Computes the number of distinct prime factors of n (small omega)
• productOfDigits(n) – Computes the product of the digits of an integer n
• radical(n) – Computes the radical of n (squarefree kernel)
• reducedTotient(n) – Returns reduced fraction φ(n)/n as { num, den }
• sigma(n) – Computes the sum of all positive divisors of n
• sigmaUnitary(n) – Computes the sum of unitary divisors σ*(n)
• sumOfDigits(n) – Computes the sum of the digits of an integer n
• tau(n) – Computes the number of positive divisors of n
• tauUnitary(n) – Computes the number of unitary divisors τ*(n)
• totient(n) – Computes Euler's Totient Function φ(n)

Prime

• countPrimes(n) – Counts the number of prime numbers ≤ n
• countPrimesInRange(a, b) – Counts the number of prime numbers between a and b (inclusive)
• cousinPrimesInRange(a, b) – Returns cousin prime pairs (p, p+4) in [a, b]
• isCoprime(a, b) – Checks whether two integers are coprime (i.e., gcd(a, b) = 1)
• isFermatPrime(n) – Checks if a number is a Fermat prime
• isLucasLehmer(p) – Lucas-Lehmer primality test for Mersenne numbers
• isMersennePrime(n) – Checks if a number is a Mersenne prime
• isMillerRabin(n, k) – Miller-Rabin probabilistic primality test
• isPrime(n) – Checks if a number is prime
• isPrimePower(n) – Returns { p, k } if n = p^k with k ≥ 2, else null
• isSafePrime(n) – Checks if a prime is a safe prime
• jacobiSymbol(a, n) – Computes the Jacobi symbol (a|n)
• largestPrimeFactor(n) – Returns the largest prime factor of n (BigInt; null if n < 2)
• legendreSymbol(a, p) – Computes the Legendre symbol (a|p)
• maxPrimeGapUpTo(n) – Largest gap between consecutive primes ≤ n as { gap, from, to }
• nextPrime(n) – Finds the smallest prime number greater than n
• nthPrime(k) – Returns the k-th prime (BigInt), k ≥ 1
• pollardRho(n) – Pollard's rho integer factorization algorithm
• prevPrime(n) – Returns the largest prime ≤ n (null if n < 2)
• primeFactorization(n) – Returns the full prime factorization of n as an array of objects { prime, power }.
• primeFactors(n) – Returns the distinct prime factors of n as an array of BigInts.
• primeGapAt(n) – Returns { p, next, gap } around n where p ≤ n and next > p
• sexyPrimesInRange(a, b) – Returns sexy prime pairs (p, p+6) in [a, b]
• sieve(n) – Generates all primes up to n using the Sieve of Eratosthenes
• sieveRange(a, b) – Generates all primes between a and b (inclusive) using the Sieve of Eratosthenes
• smallestPrimeFactor(n) – Returns the smallest prime factor of n (BigInt; null if n < 2)
• sophieGermainPrimesInRange(a, b) – Returns primes p in [a, b] where 2p + 1 is also prime
• sumOfPrimes(n) – Returns the BigInt sum of all primes ≤ n
• twinPrimeCountInRange(a, b) – Counts twin prime pairs (p, p+2) in [a, b]
• twinPrimesInRange(a, b) – Returns all twin prime pairs (p, p+2) in [a, b]

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Running Tests

npm test

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License

MIT License © 2025