numbers (version 0.8-5)

primeFactors: Prime Factors

Description

primeFactors computes a vector containing the prime factors of n. radical returns the product of those unique prime factors.

Usage

primeFactors(n)
  radical(n)

Value

Vector containing the prime factors of n, resp. the product of unique prime factors.

Arguments

n

nonnegative integer

Details

Computes the prime factors of n in ascending order, each one as often as its multiplicity requires, such that n == prod(primeFactors(n)).

## radical() is used in the abc-conjecture:

# abc-triple: 1 <= a < b, a, b coprime, c = a + b

# for every e > 0 there are only finitely many abc-triples with

# c > radical(a*b*c)^(1+e)

See Also

divisors, gmp::factorize

Examples

  primeFactors(1002001)         # 7  7  11  11  13  13
  primeFactors(65537)           # is prime
  # Euler's calculation
  primeFactors(2^32 + 1)        # 641  6700417

  radical(1002001)              # 1001

if (FALSE) {
  for (i in 1:99) {
    for (j in (i+1):100) {
      if (coprime(i, j)) {
        k = i + j
        r = radical(i*j*k)
        q = log(k) / log(r)  # 'quality' of the triple
        if (q > 1)
          cat(q, ":\t", i, ",", j, ",", k, "\n")
        }
      }
    }}