Namespace: PrimeMath

The PrimeMath namespace with static arithmetic and number theory functions Aligns with the Prime Framework by leveraging prime factorization for operations

Source: PrimeMath.js, line 116

Methods

(static) add(a, b) → {BigInt}

Add two numbers Uses regular addition for numeric types and coordinates-based addition for UniversalNumbers

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 130

Returns:

Sum of a and b

Type:: BigInt

(static) areCoherent(a, b) → {boolean}

Check if two numbers are coherent in the Prime Framework This means they represent the same abstract value despite potentially different representations

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 1501

Returns:

True if the numbers are coherent (represent the same value)

Type:: boolean

(static) coherenceDistance(a, b) → {BigInt|UniversalNumber}

Compute the distance between two numbers in the Prime Framework's fiber algebra Based on the coherence inner product and norm

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 1452

Returns:

The distance value

Type:: BigInt | UniversalNumber

(static) coherenceInnerProduct(a, b) → {BigInt|UniversalNumber}

Compute the coherence inner product between two numbers This measures their geometric alignment in the Prime Framework's fiber algebra

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 1389

Returns:

The coherence inner product value

Type:: BigInt | UniversalNumber

(static) coherenceNorm(n) → {BigInt|UniversalNumber}

Compute the coherence norm of a number in the Prime Framework Measures the "magnitude" of the number in its universal representation

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number

Source: PrimeMath.js, line 1422

Returns:

The coherence norm value

Type:: BigInt | UniversalNumber

(static) discreteLog(g, h, p, optionsopt) → {BigInt|null}

Calculate the discrete logarithm: find the smallest non-negative integer x such that g^x ≡ h (mod p) Uses Shanks' baby-step giant-step algorithm with efficiency improvements

Parameters:

Name Type Attributes Description

g

number | string | BigInt | UniversalNumber

The base

h

number | string | BigInt | UniversalNumber

The target value

p

number | string | BigInt | UniversalNumber

The modulus (should be prime for reliable results)

options

Object

Options for the algorithm

Properties

Name	Type	Attributes	Default	Description
`verify`	boolean	<optional>	true	Whether to verify the result

Source: PrimeMath.js, line 1287

Throws:

If parameters are invalid
Type PrimeMathError

Returns:

The discrete logarithm, or null if no solution exists

Type:: BigInt | null

(static) divide(a, b) → {BigInt|UniversalNumber}

Perform exact division of one number by another Only succeeds if the division is exact (no remainder) For factorized numbers, division is performed by subtracting prime exponents

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	Dividend
`b`	number \| string \| BigInt \| UniversalNumber	Divisor

Source: PrimeMath.js, line 223

Throws:

If division is not exact or divisor is zero
Type PrimeMathError

Returns:

Result of a / b

Type:: BigInt | UniversalNumber

(static) factorize(n, optionsopt) → {Map.<BigInt, BigInt>}

Factorize a number into its prime factors Provides direct access to the factorization functionality

Parameters:

Name Type Attributes Description

n

number | string | BigInt | UniversalNumber

The number to factorize

options

Object

Options for factorization

Properties

Name	Type	Attributes	Default	Description
`advanced`	boolean	<optional>	false	Whether to use advanced factorization for large numbers

Source: PrimeMath.js, line 1128

Returns:

A map where keys are prime factors and values are their exponents

Type:: Map.<BigInt, BigInt>

(static) fftMultiply(a, b) → {BigInt|UniversalNumber}

Perform a high-performance FFT-based multiplication of two large numbers Optimized for very large numbers beyond standard multiplication capabilities

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 1517

Returns:

Product of a and b

Type:: BigInt | UniversalNumber

(static) fromFactors(factors) → {BigInt|UniversalNumber}

Create a number from its prime factorization

Parameters:

Name	Type	Description
`factors`	Array.<{prime: BigInt, exponent: BigInt}> \| Map.<BigInt, BigInt>	The prime factorization

Source: PrimeMath.js, line 1144

Returns:

The number represented by the prime factorization

Type:: BigInt | UniversalNumber

(static) gcd(a, b) → {BigInt|UniversalNumber}

Find the greatest common divisor (GCD) of two numbers For factorized numbers, GCD is computed by taking the minimum of each prime's exponents

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 335

Returns:

The GCD of a and b

Type:: BigInt | UniversalNumber

(static) getDivisors(n) → {Array.<BigInt>|Array.<UniversalNumber>}

Find all divisors of a number Uses the prime factorization to efficiently generate all divisors

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to find divisors for

Source: PrimeMath.js, line 907

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

Array of all divisors of n

Type:: Array.<BigInt> | Array.<UniversalNumber>

(static) isMersennePrime(n) → {boolean}

Check if a number is a Mersenne prime (a prime of form 2^n - 1)

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to check

Source: PrimeMath.js, line 1159

Returns:

True if n is a Mersenne prime, false otherwise

Type:: boolean

(static) isPerfectNumber(n) → {boolean}

Check if a number is a perfect number (equal to the sum of its proper divisors)

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to check

Source: PrimeMath.js, line 973

Returns:

True if n is a perfect number, false otherwise

Type:: boolean

(static) isPerfectPower(n) → {Object|null}

Check if a number is a perfect power (a^b for some integers a, b with b > 1)

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to check

Source: PrimeMath.js, line 710

Returns:

Object with base and exponent if n is a perfect power, null otherwise

Type:: Object | null

(static) isPrime(n, optionsopt) → {boolean}

Check if a number is prime Uses the Miller-Rabin primality test for larger numbers For UniversalNumber, leverages the intrinsic factorization

Parameters:

Name Type Attributes Description

n

number | string | BigInt | UniversalNumber

The number to check

options

Object

Options for primality testing

Properties

Name	Type	Attributes	Default	Description
`advanced`	boolean	<optional>	true	Use advanced primality testing for large numbers

Source: PrimeMath.js, line 456

Returns:

True if n is prime, false otherwise

Type:: boolean

(static) jacobiSymbol(a, n) → {number}

Compute the Jacobi symbol (a/n) for any integer a and positive odd integer n This is a generalization of the Legendre symbol to composite moduli

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	The input number
`n`	number \| string \| BigInt \| UniversalNumber	The modulus (must be odd and positive)

Source: PrimeMath.js, line 1212

Throws:

If n is not a positive odd integer
Type PrimeMathError

Returns:

The Jacobi symbol value: 1, -1, or 0

Type:: number

(static) lcm(a, b) → {BigInt|UniversalNumber}

Find the least common multiple (LCM) of two numbers For factorized numbers, LCM is computed by taking the maximum of each prime's exponents

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 396

Returns:

The LCM of a and b

Type:: BigInt | UniversalNumber

(static) legendreSymbol(a, p) → {number}

Compute the Legendre symbol (a/p) for a number and a prime modulus This determines if a is a quadratic residue modulo p

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	The input number
`p`	number \| string \| BigInt \| UniversalNumber	The prime modulus

Source: PrimeMath.js, line 1178

Throws:

If p is not a prime number
Type PrimeMathError

Returns:

1 if a is a quadratic residue modulo p, -1 if a is a quadratic non-residue, 0 if a is divisible by p

Type:: number

(static) modInverse(a, m) → {BigInt|UniversalNumber|null}

Calculate the modular inverse of a number (a^-1 mod m)

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	The number to find the inverse for
`m`	number \| string \| BigInt \| UniversalNumber	The modulus

Source: PrimeMath.js, line 582

Throws:

If the modulus is not positive
Type PrimeMathError

Returns:

The modular inverse, or null if it doesn't exist

Type:: BigInt | UniversalNumber | null

(static) modPow(base, exponent, modulus) → {BigInt|UniversalNumber}

Perform modular exponentiation (a^b mod n)

Parameters:

Name	Type	Description
`base`	number \| string \| BigInt \| UniversalNumber	The base
`exponent`	number \| string \| BigInt \| UniversalNumber	The exponent
`modulus`	number \| string \| BigInt \| UniversalNumber	The modulus

Source: PrimeMath.js, line 651

Throws:

If modulus is not positive
Type PrimeMathError

Returns:

Result of (base^exponent) mod modulus

Type:: BigInt | UniversalNumber

(static) moebius(n) → {BigInt}

Calculate the Möbius function μ(n) value for a number Uses the prime factorization for efficient calculation

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The input number

Source: PrimeMath.js, line 869

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

The Möbius function value: 1 if n is square-free with even number of prime factors, -1 if n is square-free with odd number of primes, 0 if n has a squared prime factor

Type:: BigInt

(static) multiply(a, b) → {BigInt|UniversalNumber}

Multiply two numbers For factorized numbers, multiplication is performed by combining their prime exponent maps

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number
`b`	number \| string \| BigInt \| UniversalNumber	Second number

Source: PrimeMath.js, line 181

Returns:

Product of a and b

Type:: BigInt | UniversalNumber

(static) nextPrime(n) → {BigInt|UniversalNumber}

Find the next prime number after a given number

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The starting number

Source: PrimeMath.js, line 496

Returns:

The next prime number after n

Type:: BigInt | UniversalNumber

(static) nthPrime(n) → {BigInt|UniversalNumber}

Return the nth prime number Provides direct access to number theory sequences in the Prime Framework

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt	The index (1-based) of the prime to retrieve

Source: PrimeMath.js, line 521

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

The nth prime number

Type:: BigInt | UniversalNumber

(static) optimizeToCanonicalForm(n) → {BigInt|UniversalNumber}

Optimize a number to its canonical form in the Prime Framework Ensures the number has minimal coherence norm for its value

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to optimize

Source: PrimeMath.js, line 1474

Returns:

The canonical form with minimal norm

Type:: BigInt | UniversalNumber

(static) pow(base, exponent) → {BigInt|UniversalNumber}

Raise a number to a power For factorized numbers, exponentiation is performed by multiplying prime exponents

Parameters:

Name	Type	Description
`base`	number \| string \| BigInt \| UniversalNumber	The base
`exponent`	number \| string \| BigInt	The exponent (must be non-negative)

Source: PrimeMath.js, line 292

Throws:

If exponent is negative
Type PrimeMathError

Returns:

base^exponent

Type:: BigInt | UniversalNumber

(static) primorial(n) → {BigInt|UniversalNumber}

Compute the primorial of n (the product of all primes ≤ n)

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	Upper limit

Source: PrimeMath.js, line 544

Returns:

The primorial of n

Type:: BigInt | UniversalNumber

(static) radical(n) → {BigInt|UniversalNumber}

Calculate the radical of a number (product of distinct prime factors)

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The number to find the radical for

Source: PrimeMath.js, line 998

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

The radical of n

Type:: BigInt | UniversalNumber

(static) subtract(a, b) → {BigInt}

Subtract one number from another

Parameters:

Name	Type	Description
`a`	number \| string \| BigInt \| UniversalNumber	First number (minuend)
`b`	number \| string \| BigInt \| UniversalNumber	Second number (subtrahend)

Source: PrimeMath.js, line 152

Returns:

Difference of a - b

Type:: BigInt

(static) sumOfDivisors(n, kopt) → {BigInt|UniversalNumber}

Calculate the sum of divisors function σ(n) Uses the prime factorization for efficient calculation

Parameters:

Name	Type	Attributes	Default	Description
`n`	number \| string \| BigInt \| UniversalNumber			The number to calculate for
`k`	number \| string \| BigInt	<optional>	1	The power to raise each divisor to

Source: PrimeMath.js, line 1041

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

The sum of divisors raised to power k

Type:: BigInt | UniversalNumber

(static) totient(n) → {BigInt|UniversalNumber}

Calculate the Euler's totient function φ(n) - the count of numbers <= n that are coprime to n Uses the prime factorization for efficient calculation

Parameters:

Name	Type	Description
`n`	number \| string \| BigInt \| UniversalNumber	The input number

Source: PrimeMath.js, line 825

Throws:

If n is not a positive integer
Type PrimeMathError

Returns:

The value of φ(n)

Type:: BigInt | UniversalNumber