Module: Primes::Utils

Included in:
Integer
Defined in:
lib/primes/utils.rb,
lib/primes/utils/version.rb

Constant Summary collapse

VERSION = 
"1.1.1"

Instance Method Summary collapse

Instance Method Details

#factors(p = 13) ⇒ Object Also known as: prime_division

P13 is default prime generator here



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# File 'lib/primes/utils.rb', line 23

def factors(p=0) # p is unused dummy variable for method consistency
  factors = `factor #{self.abs}`.split(' ')[1..-1].map {|i| i.to_i}
  h = Hash.new(0); factors.each {|f| h[f] +=1}; h.to_a.sort
end

#prime? ⇒ Boolean

use pure ruby versions for platforms without cli command ‘factor’

Returns:

  • (Boolean) 


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# File 'lib/primes/utils.rb', line 32

def prime?
  `factor #{self.abs}`.split(' ').size == 2
end

#primemr?(k = 20) ⇒ Boolean

increase k for more reliability

Returns:

  • (Boolean) 


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# File 'lib/primes/utils.rb', line 227

def primemr?(k=20)  # increase k for more reliability
  n = self.abs
  return true  if n == 2 or n == 3
  return false if n % 6 != 1 && n % 6 != 5 or n == 1

  d = n - 1
  s = 0
  (d >>= 1; s += 1) while d.even?
  k.times do
    a = 2 + rand(n-4)
    x = a.to_bn.mod_exp(d,n)       #x = (a**d) mod n
    next if x == 1 or x == n-1
    (s-1).times do
      x = x.mod_exp(2,n)           #x = (x**2) mod n
      return false if x == 1
      break if x == n-1
    end
    return false if x != n-1
  end
  true  # with high probability
end

#primenth(p = 11) ⇒ Object Also known as: nthprime

return value of nth prime



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# File 'lib/primes/utils.rb', line 101

def primenth(p=11) # return value of nth prime
  # Return value of nth prime
  # Uses sozP11 Sieve of Zakiya (SoZ) as default prime sieve

  seeds  = [2, 3, 5, 7, 11, 13, 17, 19]
  primes = []

  n = self.abs                  # the desired nth prime
  return n != 0 ? seeds[n-1] : 0  if n < seeds.size

  # compute approximate value of (>) nth prime to count up to
  numb = (n*(Math.log(n)+2)).to_i

  # find primes <= Pn, compute modPn then Prime Gen residues for Pn
  primes = seeds[0..seeds.index(p)]; mod = primes.reduce(:*)
  residues=[1]; 3.step(mod,2) {|i| residues << i if mod.gcd(i) == 1}
  residues << mod+1; rescnt = residues.size-1

  num = numb-1 | 1              # if N even number then subtract 1
  k=num/mod; modk = mod*k; r=1  # kth residue group; base num value
  while num >= modk+residues[r]; r += 1 end # compute extra prms size
  maxprms = k*rescnt + r-1      # max size of prime candidates array
  prms=Array.new(maxprms,true)  # set all prime candidates to True

  # array of residues offsets to compute nonprimes positions in prms
  pos =[]; rescnt.times {|i| pos[residues[i]] = i-1};

  # sieve (SoZ) to eliminate nonprimes from prms
  sqrtN = Math.sqrt(num).to_i
  modk,r,k=0,0,0
  prms.each do |prime|
    r +=1; if r > rescnt; r=1; modk +=mod; k +=1 end
    next unless prime
    res_r = residues[r]
    prime = modk + res_r
    break if prime > sqrtN
    prmstep = prime * rescnt
    residues[1..-1].each do |ri|
      # compute (prime * (modk + ri)) position index in prms
      kk,rr  = (res_r * ri).divmod mod  # residues product res[group|track]
      nonprm = (k*(prime + ri) + kk)*rescnt + pos[rr]
      while nonprm < maxprms; prms[nonprm]=nil; nonprm +=prmstep end
    end
  end
  # the prms array now has all the primes positions for primes r1..N
  # count up to nth prime, compute value from address, and output
  prmcnt = primes.size          # number of primes up to Pn
  m=0
  while prmcnt < n; prmcnt +=1 if prms[m]; m +=1 end
  k,rr = m.divmod rescnt
  return (mod*k + residues[rr])
end

#primes(start_num = 0) ⇒ Object 



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# File 'lib/primes/utils.rb', line 156

def primes(start_num=0)
  # Find primes between a number range: end_num - start_num
  # Use as: end_num.primes(start_num) or end_num.primes
  # If start_num omitted, will find all primes <= end_num
  # If start_num > self, values are switched to continue
  # Output is an array of the primes within the given range
  # To find count of these primes do: end_num.primes(start_num).size
  # This method uses the P13 Strictly Prime (SP) Prime Generator (PG)

  num = self.abs;  start_num = start_num.abs
  num, start_num = start_num, num  if start_num > num

  primes = [2,3,5,7,11,13]      # P13 excluded primes lists
  plast  = primes.last          # last prime in primes
  return primes.select {|p| p >= start_num && p <= num} if num <= plast

  mod = primes.reduce(:*)       # P13 modulus value 2*3*5*7*11*13

  residues=[1]; plast.step(mod,2) {|i| residues << i if mod.gcd(i) == 1}
  rescnt = residues.size        # number of residues
  residues << mod+1             # to make algorithm easier

  num = num-1 | 1               # if N even number then subtract 1
  k=num/mod; modk = mod*k; r=1  # kth residue group; base num value
  while num >= modk+residues[r]; r += 1 end # compute extra prms size
  maxprms = k*rescnt + r-1      # max size of prime candidates array
  prms=Array.new(maxprms,true)  # set all prime candidates to True

  # hash of residues offsets to compute nonprimes positions in prms
  pos =[]; rescnt.times {|i| pos[residues[i]] = i-1};

  # sieve (SoZ) to eliminate nonprimes from prms
  sqrtN= Math.sqrt(num).to_i
  modk,r,k=0,0,0
  prms.each do |prime|
    r +=1; if r > rescnt; r=1; modk +=mod; k +=1 end
    next unless prime
    res_r = residues[r]
    prime = modk + res_r
    break if prime > sqrtN
    prmstep = prime * rescnt
    residues[1..-1].each do |ri|
      # compute (prime * (modk + ri)) position index in prms
      kk,rr  = (res_r * ri).divmod mod  # residues product res[group|track]
      nonprm = (k*(prime + ri) + kk)*rescnt + pos[rr] # 1st prime multiple
      while nonprm < maxprms; prms[nonprm]=nil; nonprm +=prmstep end
    end
  end
  # the prms array now has all the primes positions for primes r1..N
  primes = start_num <= plast ? primes.drop_while {|p| p < start_num} : []
  k = (start_num-2).abs/mod  # start_num's residue group value
  modk = mod*k               # start_num's mod group value
  m = rescnt*k               # number of prms elements up to kth resgroup
  r = 0                      # begin processing at start of resgroup
  while m < maxprms
    r +=1; if r > rescnt; r=1; modk +=mod end
    if prms[m]
      prime = modk + residues[r]
	  primes << prime if prime >= start_num
    end
    m +=1
  end
  primes
end