jzakiya · August 12, 2024 16:56
diff --git a/cousinprimes_ssoz.cr b/cousinprimes_ssoz.cr
 # This Crystal source file is a multiple threaded implementation to perform an
 # extremely fast Segmented Sieve of Zakiya (SSoZ) to find Cousin Primes <= N.

 # Inputs are single values N, or ranges N1 and N2, of 64-bits, 0 -- 2^64 - 1.
 # Output is the number of cousin primes <= N, or in range N1 to N2; the last
 # cousin prime value for the range; and the total time of execution.

 # Code originally developed on a System76 laptop with an Intel I7 6700HQ cpu,
 # 2.6-3.5 GHz clock, with 8 threads, and 16GB of memory. Parameter tuning
 # probably needed to optimize for other hardware systems (ARM, PowerPC, etc).

 # Compile as: $ crystal build --release -Dpreview_mt --mcpu native cousinprimes_ssoz.cr
 # To reduce binary size do: $ strip cousinprimes_ssoz
 # Thread workers default to 4, set to system max for optimum performance.
 # Single val: $ CRYSTAL_WORKERS=8 ./cousinprimes_ssoz val1
 # Range vals: $ CRYSTAL_WORKERS=8 ./cousinprimes_ssoz val1 val2
 # val1 and val2 can be entered as either: 123456789 or 123_456_789

 # Mathematical and technical basis for implementation are explained here:
 # https://www.academia.edu/37952623/The_Use_of_Prime_Generators_to_Implement_Fast_
 # Twin_Primes_Sieve_of_Zakiya_SoZ_Applications_to_Number_Theory_and_Implications_
 # for_the_Riemann_Hypotheses
 # https://www.academia.edu/7583194/The_Segmented_Sieve_of_Zakiya_SSoZ_
 # https://www.academia.edu/19786419/PRIMES-UTILS_HANDBOOK
 # https://www.academia.edu/81206391/Twin_Primes_Segmented_Sieve_of_Zakiya_SSoZ_Explained

 # This source code, and its updates, can be found here:
 # https://gist.github.com/jzakiya/0d6987ee00f3708d6cfd6daee9920bd7

 # This code is provided free and subject to copyright and terms of the
 # GNU General Public License Version 3, GPLv3, or greater.
 # License copy/terms are here: http://www.gnu.org/licenses/

 # Copyright (c) 2017-2024; Jabari Zakiya -- jzakiya at gmail dot com
 # Last update: 2024/08/12

 # Customized gcd to determine coprimality; n > m; m odd
 def coprime?(m, n)
  while m|1 != 1; t = m; m = n % m; n = t end
  m > 0
 end

 # Compute modular inverse a^-1 to base m, e.g. a*(a^-1) mod m = 1
 def modinv(a0, m0)
  return 1 if m0 == 1
  a, m = a0, m0
  x0, inv = 0, 1
  while a > 1
    inv -= (a // m) * x0
    a, m = m, a % m
    x0, inv = inv, x0
  end
  inv += m0 if inv < 0
  inv
 end

 def gen_pg_parameters(prime)
  # Create prime generator parameters for given Pn
  puts "using Prime Generator parameters for P#{prime}"
  primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]
  modpg, res_0 = 1, 0                    # compute Pn's modulus and res_0 value
  primes.each { |prm| res_0 = prm; break if prm > prime; modpg *= prm }

  rescousins = [] of Int32               # save upper cousinpair residues here
  inverses = Array.new(modpg + 2, 0)     # save Pn's residues inverses here
  rc, inc, res = 5, 2, 0                 # use P3's PGS to generate residue candidates
  midmodpg = modpg >> 1                  # mid value for modulus
  while rc < midmodpg                    # find PG's 1st half residues
    if coprime?(rc, modpg)               # if rc a residue
      mc = modpg - rc                    # create its modular complement
      inverses[rc] = modinv(rc, modpg)   # save rc and mc inverses
      inverses[mc] = modinv(mc, modpg)   # if in cousinpair save both hi residues
      rescousins << rc << mc + 4 if res + 4 == rc
      res = rc                           # save current found residue
    end
    rc += inc; inc ^= 0b110              # create next P3 sequence rc: 5 7 11 13 17 19 ...
  end
  rescousins << (midmodpg + 2); rescousins.sort!           # create pivot hi_cp
  inverses[modpg + 1] = 1; inverses[modpg - 1] = modpg - 1 # last 2 residues are self inverses
  {modpg, res_0, rescousins.size, rescousins, inverses}
 end

 def set_sieve_parameters(start_num, end_num)
  # Select at runtime best PG and segment size parameters for input values.
  # These are good estimates derived from PG data profiling. Can be improved.
  nrange = end_num - start_num
  bn, pg = 0, 3
  if end_num < 49
    bn = 1; pg = 3
  elsif nrange < 77_000_000
    bn = 16; pg = 5
  elsif nrange <  1_100_000_000
    bn = 32; pg = 7
  elsif nrange < 35_500_000_000
    bn = 64; pg = 11
  elsif nrange < 14_000_000_000_000
    pg = 13
    if    nrange > 7_000_000_000_000; bn = 384
    elsif nrange > 2_500_000_000_000; bn = 320
    elsif nrange >   250_000_000_000; bn = 196
    else  bn = 128
    end
  else
    bn = 384; pg = 17
  end
  modpg, res_0, pairscnt, rescousins, resinvrs = gen_pg_parameters(pg)
  kmin = (start_num-2) // modpg + 1      # number of resgroups to start_num
  kmax = (end_num - 2) // modpg + 1      # number of resgroups to end_num
  krange = kmax - kmin + 1               # number of resgroups in range, at least 1
  n = krange < 37_500_000_000_000 ? 10 : (krange < 975_000_000_000_000 ? 16 : 20)
  b = bn * 1024 * n                      # set seg size to optimize for selected PG
  ks = krange < b ? krange : b           # segments resgroups size

  puts "segment size = #{ks.format} resgroups; seg array is [1 x #{(((ks-1) >> 6) + 1).format}] 64-bits"
  maxpairs = krange * pairscnt           # maximum number of twinprime pcs
  puts "twinprime candidates = #{maxpairs.format}; resgroups = #{krange.format}"
  {modpg, res_0, ks, kmin, kmax, krange, pairscnt, rescousins, resinvrs}
 end

 def sozp5(val, res_0, start_num, end_num)
  # Return the primes r0..sqrt(end_num) within range (start_num...end_num)
  md, rescnt = 30u64, 8                  # P5's modulus and residues count
  res  = [7,11,13,17,19,23,29,31]        # P5's residues
  range_size = end_num - start_num       # integers size of inputs range

  kmax = (val - 2) // md + 1             # number of resgroups upto input value
  prms = Array(UInt8).new(kmax, 0)       # byte array of prime candidates, init '0'
  sqrtn = Math.isqrt(val-1|1)            # integer sqrt of largest odd value <= val
  k = sqrtn//md; resk = sqrtn-md*k; r=0  # sqrtn resgroup|residue values; 1st res posn
  while resk >= res[r]; r &+= 1 end      # find largest residue <= sqrtn posn in its resgroup
  pcs_to_sqrtn = k &* rescnt &+ r        # number of pcs <= sqrtn

  pcs_to_sqrtn.times do |i|              # for r0..sqrtN primes mark their multiples
    k, r  = i.divmod rescnt              # update resgroup parameters
    next if prms[k] & (1 << r) != 0      # skip pc if not prime
    prm_r = res[r]                       # if prime save its residue value
    prime = md &* k &+ prm_r             # numerate its prime value
    rem = start_num % prime              # prime's modular distance to start_num
    next unless (prime - rem <= range_size) || rem == 0 # skip prime if no multiple in range
    res.each do |ri|                     # mark prime's multiples in prms
      kn,rn = (prm_r &* ri &- 2).divmod md # cross-product resgroup|residue
      bit_r = 1 << res.index(rn &+ 2).not_nil! # bit mask value for prod's residue
      kpm = k &* (prime &+ ri) &+ kn     # resgroup for prime's 1st multiple
      while kpm < kmax; prms[kpm] |= bit_r; kpm &+= prime end  # mark primes's multiples
  end end
  # prms now contains the prime multiples positions marked for the pcs r0..N
  # now along each restrack, identify|numerate the primes in each resgroup
  # return only the primes with a multiple within range (start_num...end_num)
  primes = [] of UInt64                  # create empty dynamic array for primes
  res.each_with_index do |r_i, i|        # along each restrack|row til end
    kmax.times do |k|                    # for each resgroup along restrack
      if prms[k] & (1 << i) == 0         # if bit location a prime
        prime = md &* k &+ r_i           # numerate its value, store if in range
        # check if prime has multiple in range, if so keep it, if not don't
        rem = start_num % prime          # if rem 0 then start_num is multiple of prime
        primes << prime if (res_0 <= prime <= val) && (prime &- rem <= range_size || rem == 0)
  end end end
  primes                                 # primes stored in restrack|row sequence order
 end

 def nextp_init(rhi, kmin, modpg, primes, resinvrs)
  # Initialize 'nextp' array for cousinpair upper residue rhi in 'rescousins'.
  # Compute 1st prime multiple resgroups for each prime r0..sqrt(N) and
  # store consecutively as lo_cp|hi_cp pairs for their restracks.
  nextp = Slice(UInt64).new(primes.size*2) # 1st mults array for cousinpair
  r_hi, r_lo = rhi, rhi &- 4             # upper|lower cousinpair residue values
  primes.each_with_index do |prime, j|   # for each prime r0..sqrt(N)
    k = (prime &- 2) // modpg            # find the resgroup it's in
    r = (prime &- 2) %  modpg &+ 2       # and its residue value
    r_inv = resinvrs[r].to_u64           # and residue inverse
    rl = (r_inv &* r_lo &- 2) % modpg &+ 2 # compute r's ri for r_lo
    rh = (r_inv &* r_hi &- 2) % modpg &+ 2 # compute r's ri for r_hi
    kl = k &* (prime &+ rl) &+ (r &* rl &- 2) // modpg # kl 1st mult resgroup
    kh = k &* (prime &+ rh) &+ (r &* rh &- 2) // modpg # kh 1st mult resgroup
    kl < kmin ? (kl = (kmin &- kl) % prime; kl = prime &- kl if kl > 0) : (kl &-= kmin)
    kh < kmin ? (kh = (kmin &- kh) % prime; kh = prime &- kh if kh > 0) : (kh &-= kmin)
    nextp[j * 2] = kl.to_u64             # prime's 1st mult lo_tp resgroup val in range 
    nextp[j * 2 | 1] = kh.to_u64         # prime's 1st mult hi_tp resgroup val in range
  end
  nextp
 end

 def cousins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
  # Perform in thread the ssoz for given cousinpair residues for kmax resgroups.
  # First create|init 'nextp' array of 1st prime mults for given cousinpair,
  # stored consequtively in 'nextp', and init seg array for ks resgroups.
  # For sieve, mark resgroup bits to '1' if either cousinpair restrack is nonprime
  # for primes mults resgroups, and update 'nextp' restrack slices acccordingly.
  # Return the last cousinprime|sum for the range for this cousinpair residues.
  s = 6                                                # shift value for 64 bits
  bmask = (1 << s) &- 1                                # bitmask val for 64 bits
  sum, ki, kn  = 0_u64, kmin &- 1, ks                  # init these parameters
  hi_cp, k_max = 0_u64, kmax                           # max cousinprime|resgroup
  seg = Slice(UInt64).new(((ks - 1) >> s) &+ 1)        # seg array of ks resgroups
  ki    &+= 1 if r_hi &- 4 < (start_num &- 2) % modpg &+ 2 # ensure lo cp in range
  k_max &-= 1 if r_hi > (end_num &- 2) % modpg &+ 2    # ensure hi cp in range
  nextp = nextp_init(r_hi, ki, modpg, primes,resinvrs) # init nextp array
  while ki < k_max                       # for ks size slices upto kmax
    kn = k_max &- ki if ks > (k_max &- ki) # adjust kn size for last seg
    primes.each_with_index do |prime, j| # for each prime r0..sqrt(N)
                                         # for lower cousinpair residue track
      k1 = nextp.to_unsafe[j * 2]        # starting from this resgroup in seg
      while k1 < kn                      # mark primenth resgroup bits prime mults
        seg.to_unsafe[k1 >> s] |= 1u64 << (k1 & bmask)
        k1 &+= prime  end                # set resgroup for prime's next multiple
      nextp.to_unsafe[j * 2] = k1 &- kn  # save 1st resgroup in next eligible seg
                                         # for upper cousinpair residue track
      k2 = nextp.to_unsafe[j * 2 | 1]    # starting from this resgroup in seg
      while k2 < kn                      # mark primenth resgroup bits prime mults
        seg.to_unsafe[k2 >> s] |= 1u64 << (k2 & bmask)
        k2 &+= prime  end                # set resgroup for prime's next multiple
      nextp.to_unsafe[j * 2 | 1]=k2 &- kn# save 1st resgroup in next eligible seg
    end                                  # set as nonprime unused bits in last seg[n]
                                         # so fast, do for every seg[i]
    seg.to_unsafe[(kn - 1) >> s] |= ~1u64 << ((kn &- 1) & bmask)
    cnt = 0                              # count the cousinprimes in the segment
    seg[0..(kn - 1) >> s].each { |m| cnt &+= (~m).popcount }
    if cnt > 0                           # if segment has cousinprimes
      sum &+= cnt                        # add segment count to total range count
      upk = kn &- 1                      # from end of seg, count back to largest cp
      while seg.to_unsafe[upk >> s] & (1u64 << (upk & bmask)) != 0; upk &-= 1 end
      hi_cp = ki &+ upk                  # set its full range resgroup value
    end
    ki &+= ks                            # set 1st resgroup val of next seg slice
    seg.fill(0) if ki < k_max            # set next seg to all primes if not last
  end                                    # when sieve done, numerate largest cousin
                                         # for ranges w/o cousins set largest to 1
  hi_cp = (r_hi > end_num || sum == 0) ? 0 : hi_cp &* modpg &+ r_hi
  {hi_cp.to_u64, sum.to_u64}             # return largest cousinprime|cousins count
 end

 def cousinprimes_ssoz()
  end_num   = {(ARGV[0].to_u64 underscore: true), 3u64}.max
  start_num = ARGV.size > 1 ? {(ARGV[1].to_u64 underscore: true), 3u64}.max : 3u64
  start_num, end_num = end_num, start_num if start_num > end_num
  start_num |= 1                         # if start_num even increase by 1
  end_num = (end_num - 1) | 1            # if end_num even decrease by 1
  start_num = end_num = 7 if end_num - start_num < 4

  puts "threads = #{System.cpu_count}"
  ts = Time.monotonic                    # start timing sieve setup execution
                                         # select Pn, set sieving params for inputs
  modpg, res_0, ks, kmin, kmax, krange, pairscnt, rescousins, resinvrs = set_sieve_parameters(start_num, end_num)

  # create sieve primes <= sqrt(end_num), only use those whose multiples within inputs range
  primes = end_num < 49 ? [5] : sozp5(Math.isqrt(end_num), res_0, start_num, end_num)

  puts "each of #{pairscnt.format} threads has nextp[2 x #{primes.size.format}] array"

  te = (Time.monotonic - ts).total_seconds.round(6)
  puts "setup time = #{te} secs"         # display sieve setup time
  puts "perform cousinprimes ssoz sieve"
  t1 = Time.monotonic                    # start timing ssoz sieve execution

  cnts = Array(UInt64).new(pairscnt, 0)  # cousins count for each cousinpair
  lastcousins = Array(UInt64).new(pairscnt, 0) # largest hi_cp for each cousinpair
  done = Channel(Nil).new(pairscnt)
  
  threadscnt = Atomic.new(0)             # count of finished threads
  rescousins.each_with_index do |r_hi,i| # sieve cousinpair restracks10
    spawn do
      lastcousins[i], cnts[i] = cousins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
      print "\r#{threadscnt.add(1)} of #{pairscnt} cousinpairs done"
      done.send(nil)
  end end
  pairscnt.times { done.receive }        # wait for all threads to finish
  print "\r#{pairscnt} of #{pairscnt} cousinpairs done"

  cousinscnt  = 0u64                     # count of cousinprimes in range
  last_cousin = 0u64                     # largest hi_cp cousin in range
  if end_num < 49                        # check for cousins in low ranges
    small_cousins = [11, 17, 23, 41, 47 ].select { |c_hi| start_num <= c_hi - 4 && c_hi <= end_num }
    cousinscnt, last_cousin = small_cousins.size, small_cousins.last? || 0
  else                                   # check for cousins from sieve
    cousinscnt += cnts.sum               # compute number of cousinprimes in range
    last_cousin = lastcousins.max        # find largest hi_cp cousinprime in range
  end
                                         # account for 1st cousin prime, defined as (3, 7)
  if start_num == 3 && end_num > 6; cousinscnt += 1; last_cousin = 7 if end_num < 11 end
  kn = krange % ks                       # set number of resgroups in last slice
  kn = ks if kn == 0                     # if multiple of seg size set to seg size
  t2 = (Time.monotonic - t1).total_seconds  # sieve execution time

  puts "\nsieve time = #{t2.round(6)} secs"      # ssoz sieve time
  puts "total time = #{(t2 + te).round(6)} secs" # setup + sieve time
  puts "last segment = #{kn.format} resgroups; segment slices = #{((krange - 1)//ks + 1).format}"
  puts "total cousins = #{cousinscnt.format}; last cousin = #{last_cousin.format}|-4"
 end

 cousinprimes_ssoz
	# This Crystal source file is a multiple threaded implementation to perform an
	# extremely fast Segmented Sieve of Zakiya (SSoZ) to find Cousin Primes <= N.

	# Inputs are single values N, or ranges N1 and N2, of 64-bits, 0 -- 2^64 - 1.
	# Output is the number of cousin primes <= N, or in range N1 to N2; the last
	# cousin prime value for the range; and the total time of execution.

	# Code originally developed on a System76 laptop with an Intel I7 6700HQ cpu,
	# 2.6-3.5 GHz clock, with 8 threads, and 16GB of memory. Parameter tuning
	# probably needed to optimize for other hardware systems (ARM, PowerPC, etc).

	# Compile as: $ crystal build --release -Dpreview_mt --mcpu native cousinprimes_ssoz.cr
	# To reduce binary size do: $ strip cousinprimes_ssoz
	# Thread workers default to 4, set to system max for optimum performance.
	# Single val: $ CRYSTAL_WORKERS=8 ./cousinprimes_ssoz val1
	# Range vals: $ CRYSTAL_WORKERS=8 ./cousinprimes_ssoz val1 val2
	# val1 and val2 can be entered as either: 123456789 or 123_456_789

	# Mathematical and technical basis for implementation are explained here:
	# https://www.academia.edu/37952623/The_Use_of_Prime_Generators_to_Implement_Fast_
	# Twin_Primes_Sieve_of_Zakiya_SoZ_Applications_to_Number_Theory_and_Implications_
	# for_the_Riemann_Hypotheses
	# https://www.academia.edu/7583194/The_Segmented_Sieve_of_Zakiya_SSoZ_
	# https://www.academia.edu/19786419/PRIMES-UTILS_HANDBOOK
	# https://www.academia.edu/81206391/Twin_Primes_Segmented_Sieve_of_Zakiya_SSoZ_Explained

	# This source code, and its updates, can be found here:
	# https://gist.github.com/jzakiya/0d6987ee00f3708d6cfd6daee9920bd7

	# This code is provided free and subject to copyright and terms of the
	# GNU General Public License Version 3, GPLv3, or greater.
	# License copy/terms are here: http://www.gnu.org/licenses/

	# Copyright (c) 2017-2024; Jabari Zakiya -- jzakiya at gmail dot com
	# Last update: 2024/08/12

	# Customized gcd to determine coprimality; n > m; m odd
	def coprime?(m, n)
	while m\|1 != 1; t = m; m = n % m; n = t end
	m > 0
	end

	# Compute modular inverse a^-1 to base m, e.g. a*(a^-1) mod m = 1
	def modinv(a0, m0)
	return 1 if m0 == 1
	a, m = a0, m0
	x0, inv = 0, 1
	while a > 1
	inv -= (a // m) * x0
	a, m = m, a % m
	x0, inv = inv, x0
	end
	inv += m0 if inv < 0
	inv
	end

	def gen_pg_parameters(prime)
	# Create prime generator parameters for given Pn
	puts "using Prime Generator parameters for P#{prime}"
	primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]
	modpg, res_0 = 1, 0 # compute Pn's modulus and res_0 value
	primes.each { \|prm\| res_0 = prm; break if prm > prime; modpg *= prm }

	rescousins = [] of Int32 # save upper cousinpair residues here
	inverses = Array.new(modpg + 2, 0) # save Pn's residues inverses here
	rc, inc, res = 5, 2, 0 # use P3's PGS to generate residue candidates
	midmodpg = modpg >> 1 # mid value for modulus
	while rc < midmodpg # find PG's 1st half residues
	if coprime?(rc, modpg) # if rc a residue
	mc = modpg - rc # create its modular complement
	inverses[rc] = modinv(rc, modpg) # save rc and mc inverses
	inverses[mc] = modinv(mc, modpg) # if in cousinpair save both hi residues
	rescousins << rc << mc + 4 if res + 4 == rc
	res = rc # save current found residue
	end
	rc += inc; inc ^= 0b110 # create next P3 sequence rc: 5 7 11 13 17 19 ...
	end
	rescousins << (midmodpg + 2); rescousins.sort! # create pivot hi_cp
	inverses[modpg + 1] = 1; inverses[modpg - 1] = modpg - 1 # last 2 residues are self inverses
	{modpg, res_0, rescousins.size, rescousins, inverses}
	end

	def set_sieve_parameters(start_num, end_num)
	# Select at runtime best PG and segment size parameters for input values.
	# These are good estimates derived from PG data profiling. Can be improved.
	nrange = end_num - start_num
	bn, pg = 0, 3
	if end_num < 49
	bn = 1; pg = 3
	elsif nrange < 77_000_000
	bn = 16; pg = 5
	elsif nrange < 1_100_000_000
	bn = 32; pg = 7
	elsif nrange < 35_500_000_000
	bn = 64; pg = 11
	elsif nrange < 14_000_000_000_000
	pg = 13
	if nrange > 7_000_000_000_000; bn = 384
	elsif nrange > 2_500_000_000_000; bn = 320
	elsif nrange > 250_000_000_000; bn = 196
	else bn = 128
	end
	else
	bn = 384; pg = 17
	end
	modpg, res_0, pairscnt, rescousins, resinvrs = gen_pg_parameters(pg)
	kmin = (start_num-2) // modpg + 1 # number of resgroups to start_num
	kmax = (end_num - 2) // modpg + 1 # number of resgroups to end_num
	krange = kmax - kmin + 1 # number of resgroups in range, at least 1
	n = krange < 37_500_000_000_000 ? 10 : (krange < 975_000_000_000_000 ? 16 : 20)
	b = bn * 1024 * n # set seg size to optimize for selected PG
	ks = krange < b ? krange : b # segments resgroups size

	puts "segment size = #{ks.format} resgroups; seg array is [1 x #{(((ks-1) >> 6) + 1).format}] 64-bits"
	maxpairs = krange * pairscnt # maximum number of twinprime pcs
	puts "twinprime candidates = #{maxpairs.format}; resgroups = #{krange.format}"
	{modpg, res_0, ks, kmin, kmax, krange, pairscnt, rescousins, resinvrs}
	end

	def sozp5(val, res_0, start_num, end_num)
	# Return the primes r0..sqrt(end_num) within range (start_num...end_num)
	md, rescnt = 30u64, 8 # P5's modulus and residues count
	res = [7,11,13,17,19,23,29,31] # P5's residues
	range_size = end_num - start_num # integers size of inputs range

	kmax = (val - 2) // md + 1 # number of resgroups upto input value
	prms = Array(UInt8).new(kmax, 0) # byte array of prime candidates, init '0'
	sqrtn = Math.isqrt(val-1\|1) # integer sqrt of largest odd value <= val
	k = sqrtn//md; resk = sqrtn-md*k; r=0 # sqrtn resgroup\|residue values; 1st res posn
	while resk >= res[r]; r &+= 1 end # find largest residue <= sqrtn posn in its resgroup
	pcs_to_sqrtn = k &* rescnt &+ r # number of pcs <= sqrtn

	pcs_to_sqrtn.times do \|i\| # for r0..sqrtN primes mark their multiples
	k, r = i.divmod rescnt # update resgroup parameters
	next if prms[k] & (1 << r) != 0 # skip pc if not prime
	prm_r = res[r] # if prime save its residue value
	prime = md &* k &+ prm_r # numerate its prime value
	rem = start_num % prime # prime's modular distance to start_num
	next unless (prime - rem <= range_size) \|\| rem == 0 # skip prime if no multiple in range
	res.each do \|ri\| # mark prime's multiples in prms
	kn,rn = (prm_r &* ri &- 2).divmod md # cross-product resgroup\|residue
	bit_r = 1 << res.index(rn &+ 2).not_nil! # bit mask value for prod's residue
	kpm = k &* (prime &+ ri) &+ kn # resgroup for prime's 1st multiple
	while kpm < kmax; prms[kpm] \|= bit_r; kpm &+= prime end # mark primes's multiples
	end end
	# prms now contains the prime multiples positions marked for the pcs r0..N
	# now along each restrack, identify\|numerate the primes in each resgroup
	# return only the primes with a multiple within range (start_num...end_num)
	primes = [] of UInt64 # create empty dynamic array for primes
	res.each_with_index do \|r_i, i\| # along each restrack\|row til end
	kmax.times do \|k\| # for each resgroup along restrack
	if prms[k] & (1 << i) == 0 # if bit location a prime
	prime = md &* k &+ r_i # numerate its value, store if in range
	# check if prime has multiple in range, if so keep it, if not don't
	rem = start_num % prime # if rem 0 then start_num is multiple of prime
	primes << prime if (res_0 <= prime <= val) && (prime &- rem <= range_size \|\| rem == 0)
	end end end
	primes # primes stored in restrack\|row sequence order
	end

	def nextp_init(rhi, kmin, modpg, primes, resinvrs)
	# Initialize 'nextp' array for cousinpair upper residue rhi in 'rescousins'.
	# Compute 1st prime multiple resgroups for each prime r0..sqrt(N) and
	# store consecutively as lo_cp\|hi_cp pairs for their restracks.
	nextp = Slice(UInt64).new(primes.size*2) # 1st mults array for cousinpair
	r_hi, r_lo = rhi, rhi &- 4 # upper\|lower cousinpair residue values
	primes.each_with_index do \|prime, j\| # for each prime r0..sqrt(N)
	k = (prime &- 2) // modpg # find the resgroup it's in
	r = (prime &- 2) % modpg &+ 2 # and its residue value
	r_inv = resinvrs[r].to_u64 # and residue inverse
	rl = (r_inv &* r_lo &- 2) % modpg &+ 2 # compute r's ri for r_lo
	rh = (r_inv &* r_hi &- 2) % modpg &+ 2 # compute r's ri for r_hi
	kl = k &* (prime &+ rl) &+ (r &* rl &- 2) // modpg # kl 1st mult resgroup
	kh = k &* (prime &+ rh) &+ (r &* rh &- 2) // modpg # kh 1st mult resgroup
	kl < kmin ? (kl = (kmin &- kl) % prime; kl = prime &- kl if kl > 0) : (kl &-= kmin)
	kh < kmin ? (kh = (kmin &- kh) % prime; kh = prime &- kh if kh > 0) : (kh &-= kmin)
	nextp[j * 2] = kl.to_u64 # prime's 1st mult lo_tp resgroup val in range
	nextp[j * 2 \| 1] = kh.to_u64 # prime's 1st mult hi_tp resgroup val in range
	end
	nextp
	end

	def cousins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
	# Perform in thread the ssoz for given cousinpair residues for kmax resgroups.
	# First create\|init 'nextp' array of 1st prime mults for given cousinpair,
	# stored consequtively in 'nextp', and init seg array for ks resgroups.
	# For sieve, mark resgroup bits to '1' if either cousinpair restrack is nonprime
	# for primes mults resgroups, and update 'nextp' restrack slices acccordingly.
	# Return the last cousinprime\|sum for the range for this cousinpair residues.
	s = 6 # shift value for 64 bits
	bmask = (1 << s) &- 1 # bitmask val for 64 bits
	sum, ki, kn = 0_u64, kmin &- 1, ks # init these parameters
	hi_cp, k_max = 0_u64, kmax # max cousinprime\|resgroup
	seg = Slice(UInt64).new(((ks - 1) >> s) &+ 1) # seg array of ks resgroups
	ki &+= 1 if r_hi &- 4 < (start_num &- 2) % modpg &+ 2 # ensure lo cp in range
	k_max &-= 1 if r_hi > (end_num &- 2) % modpg &+ 2 # ensure hi cp in range
	nextp = nextp_init(r_hi, ki, modpg, primes,resinvrs) # init nextp array
	while ki < k_max # for ks size slices upto kmax
	kn = k_max &- ki if ks > (k_max &- ki) # adjust kn size for last seg
	primes.each_with_index do \|prime, j\| # for each prime r0..sqrt(N)
	# for lower cousinpair residue track
	k1 = nextp.to_unsafe[j * 2] # starting from this resgroup in seg
	while k1 < kn # mark primenth resgroup bits prime mults
	seg.to_unsafe[k1 >> s] \|= 1u64 << (k1 & bmask)
	k1 &+= prime end # set resgroup for prime's next multiple
	nextp.to_unsafe[j * 2] = k1 &- kn # save 1st resgroup in next eligible seg
	# for upper cousinpair residue track
	k2 = nextp.to_unsafe[j * 2 \| 1] # starting from this resgroup in seg
	while k2 < kn # mark primenth resgroup bits prime mults
	seg.to_unsafe[k2 >> s] \|= 1u64 << (k2 & bmask)
	k2 &+= prime end # set resgroup for prime's next multiple
	nextp.to_unsafe[j * 2 \| 1]=k2 &- kn# save 1st resgroup in next eligible seg
	end # set as nonprime unused bits in last seg[n]
	# so fast, do for every seg[i]
	seg.to_unsafe[(kn - 1) >> s] \|= ~1u64 << ((kn &- 1) & bmask)
	cnt = 0 # count the cousinprimes in the segment
	seg[0..(kn - 1) >> s].each { \|m\| cnt &+= (~m).popcount }
	if cnt > 0 # if segment has cousinprimes
	sum &+= cnt # add segment count to total range count
	upk = kn &- 1 # from end of seg, count back to largest cp
	while seg.to_unsafe[upk >> s] & (1u64 << (upk & bmask)) != 0; upk &-= 1 end
	hi_cp = ki &+ upk # set its full range resgroup value
	end
	ki &+= ks # set 1st resgroup val of next seg slice
	seg.fill(0) if ki < k_max # set next seg to all primes if not last
	end # when sieve done, numerate largest cousin
	# for ranges w/o cousins set largest to 1
	hi_cp = (r_hi > end_num \|\| sum == 0) ? 0 : hi_cp &* modpg &+ r_hi
	{hi_cp.to_u64, sum.to_u64} # return largest cousinprime\|cousins count
	end

	def cousinprimes_ssoz()
	end_num = {(ARGV[0].to_u64 underscore: true), 3u64}.max
	start_num = ARGV.size > 1 ? {(ARGV[1].to_u64 underscore: true), 3u64}.max : 3u64
	start_num, end_num = end_num, start_num if start_num > end_num
	start_num \|= 1 # if start_num even increase by 1
	end_num = (end_num - 1) \| 1 # if end_num even decrease by 1
	start_num = end_num = 7 if end_num - start_num < 4

	puts "threads = #{System.cpu_count}"
	ts = Time.monotonic # start timing sieve setup execution
	# select Pn, set sieving params for inputs
	modpg, res_0, ks, kmin, kmax, krange, pairscnt, rescousins, resinvrs = set_sieve_parameters(start_num, end_num)

	# create sieve primes <= sqrt(end_num), only use those whose multiples within inputs range
	primes = end_num < 49 ? [5] : sozp5(Math.isqrt(end_num), res_0, start_num, end_num)

	puts "each of #{pairscnt.format} threads has nextp[2 x #{primes.size.format}] array"

	te = (Time.monotonic - ts).total_seconds.round(6)
	puts "setup time = #{te} secs" # display sieve setup time
	puts "perform cousinprimes ssoz sieve"
	t1 = Time.monotonic # start timing ssoz sieve execution

	cnts = Array(UInt64).new(pairscnt, 0) # cousins count for each cousinpair
	lastcousins = Array(UInt64).new(pairscnt, 0) # largest hi_cp for each cousinpair
	done = Channel(Nil).new(pairscnt)

	threadscnt = Atomic.new(0) # count of finished threads
	rescousins.each_with_index do \|r_hi,i\| # sieve cousinpair restracks10
	spawn do
	lastcousins[i], cnts[i] = cousins_sieve(r_hi, kmin, kmax, ks, start_num, end_num, modpg, primes, resinvrs)
	print "\r#{threadscnt.add(1)} of #{pairscnt} cousinpairs done"
	done.send(nil)
	end end
	pairscnt.times { done.receive } # wait for all threads to finish
	print "\r#{pairscnt} of #{pairscnt} cousinpairs done"

	cousinscnt = 0u64 # count of cousinprimes in range
	last_cousin = 0u64 # largest hi_cp cousin in range
	if end_num < 49 # check for cousins in low ranges
	small_cousins = [11, 17, 23, 41, 47 ].select { \|c_hi\| start_num <= c_hi - 4 && c_hi <= end_num }
	cousinscnt, last_cousin = small_cousins.size, small_cousins.last? \|\| 0
	else # check for cousins from sieve
	cousinscnt += cnts.sum # compute number of cousinprimes in range
	last_cousin = lastcousins.max # find largest hi_cp cousinprime in range
	end
	# account for 1st cousin prime, defined as (3, 7)
	if start_num == 3 && end_num > 6; cousinscnt += 1; last_cousin = 7 if end_num < 11 end
	kn = krange % ks # set number of resgroups in last slice
	kn = ks if kn == 0 # if multiple of seg size set to seg size
	t2 = (Time.monotonic - t1).total_seconds # sieve execution time

	puts "\nsieve time = #{t2.round(6)} secs" # ssoz sieve time
	puts "total time = #{(t2 + te).round(6)} secs" # setup + sieve time
	puts "last segment = #{kn.format} resgroups; segment slices = #{((krange - 1)//ks + 1).format}"
	puts "total cousins = #{cousinscnt.format}; last cousin = #{last_cousin.format}\|-4"
	end

	cousinprimes_ssoz