jzakiya · August 12, 2024 17:06 · Pascal66 · Dec 17, 2019
diff --git a/twinprimes_ssoz.nim b/twinprimes_ssoz.nim
 #[
 This Nim source file is a multiple threaded implementation to perform an
 extremely fast Segmented Sieve of Zakiya (SSoZ) to find Twin Primes <= N.
 
 Inputs are single values N, or ranges N1 and N2, of 64-bits, 0 -- 2^64 - 1.
 Output is the number of twin primes <= N, or in range N1 to N2; the last
 twin 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
 would be needed to optimize for other hardware systems (ARM, PowerPC, etc).

 For Nim >= 2.0.0 compile as:
 $ nim c --cc:gcc --mm:arc --d:danger --d:release --threads:on --d:useMalloc twinprimes_ssozgist_new.nim
 
 Then run executable: $ ./twinprimes_ssozgist1 <cr>, and enter range values.
 Or alternatively: $ echo <range1> <range2> | ./twinprimes_ssoz
 Range values can be provided in either order (larger or smaller first).
 
 This source file, and updates, will be available here:
 https://gist.github.com/jzakiya/6c7e1868bd749a6b1add62e3e3b2341e
 
 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 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
 Version Date: 2024/08/12
 ]#

 import math                              # for sqrt, gcd, mod functions
 import strutils, typetraits              # for number input
 import times, os                         # for timing code execution
 import osproc                            # for getting threads count
 import threadpool                        # for parallel processing
 import algorithm                         # for sort
 import bitops                            # for countSetBits
 {.experimental: "parallel".}             # required to use 'parallel' (>= 1.4.x)

 proc modinv(a0, b0: int): int =
  ## Compute modular inverse a^-1 of a to base b, e.g. a*(a^-1) mod b = 1.
  var (a, b, x0) = (a0, b0, 0)
  result = 1
  if b == 1: return
  while a > 1:
    result -= (a div b) * x0
    a = a mod b
    swap a, b
    swap x0, result
  if result < 0: result += b0

 proc genPGparameters(prime: int): (int, int, int, seq[int], seq[int]) =
  ## Create prime generator parameters for given Pn
  echo("using Prime Generator parameters for P", prime)
  let primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]
  var (modpg, res_0) = (1, 0)            # compute Pn's modulus and res_0 value
  for prm in primes: (res_0 = prm; if prm > prime: break; modpg *= prm)

  var restwins: seq[int] = @[]           # save upper twin pair residues here
  var inverses = newSeq[int](modpg+2)    # save Pn's residues inverses here
  var (rc, inc, res) = (5, 2, 0)         # use P3's PGS to generate residue candidates
  while rc < (modpg shr 1):              # find PG's 1st half residues
    if gcd(modpg, rc) == 1:              # if rc is a residue
      let mc = modpg - rc                # create its modular complement
      inverses[rc] = modinv(rc,modpg)    # save rc amd mc inverses
      inverses[mc] = modinv(mc,modpg)    # if in twinpair save both hi residues
      if res + 2 == rc: restwins.add(rc); restwins.add(mc + 2)
      res = rc                           # save current found residue
    rc += inc; inc = inc xor 0b110       # create next P3 sequence rc: 5 7 11 13 17 19 ...

  restwins.sort(system.cmp[int]); restwins.add(modpg + 1)   # last residue is last hi_tp
  inverses[modpg + 1] = 1;  inverses[modpg - 1] = modpg - 1 # last 2 resdiues are self inverses
  (modpg, res_0, restwins.len, restwins, inverses)

 # Global parameters
 var
  cnts:      seq[uint]                   # holds twinprimes count for each twinpair
  lastwins:  seq[uint]                   # holds largest twinprime for each twinpair

 proc setSieveParameters(start_num, end_num: uint): (uint, int, uint, uint, uint, uint, int, seq[int], seq[int]) =
  ## Select at runtime best PG and segment size factor to use for input value.
  ## These are good estimates derived from PG data profiling. Can be improved.
  let range = end_num - start_num
  var (Bn, pg) = (0, 3)
  if end_num < 49'u:
    Bn = 1; pg = 3
  elif range < 77_000_000:
    Bn = 16; pg = 5
  elif range < 1_100_000_000'u:
    Bn = 32; pg = 7
  elif range < 35_500_000_000'u:
    Bn = 64; pg = 11
  elif range < 14_000_000_000_000'u:
    pg = 13
    if   range > 7_000_000_000_000'u: Bn = 384
    elif range > 2_500_000_000_000'u: Bn = 320
    elif range >   250_000_000_000'u: Bn = 196
    else: Bn = 128
  else:
    Bn = 384; pg = 17
  let (modpg, res_0, pairscnt, restwins, resinvrs) = genPGparameters(pg)
  let Kmin = (start_num - 2) div modpg.uint + 1 # number of resgroups to start_num
  let Kmax = (end_num - 2)   div modpg.uint + 1 # number of resgroups to end_num
  let krange = Kmax - Kmin + 1                  # number of range resgroups, at least 1
  let n = if krange < 37_500_000_000_000'u: 10 elif krange < 975_000_000_000_000'u: 16 else: 20
  let B = uint(Bn * 1024 * n)                   # set seg size to optimize for selected PG
  let Ks = if krange < B: krange else: B        # segments resgroups size

  echo("segment size = ",Ks," resgroups; seg array is [1 x ",((Ks-1) shr 6) + 1,"] 64-bits")
  let maxpairs = krange.int * pairscnt;        # maximum number of twinprime pcs
  echo("twinprime candidates = ", maxpairs, "; resgroups = ", krange);
  (modpg.uint, res_0, Ks, Kmin, Kmax, krange, pairscnt, restwins, resinvrs)

 proc sozp5(val: int | int64, res_0: int, start_num, end_num: uint): seq[int] =
  ## Return the primes r0..sqrt(end_num) within range (start_num...end_num)
  let (md, rescnt) = (30, 8)             # P5's modulus and residues count
  let res = [7,11,13,17,19,23,29,31]     # P5's residues list
  let bitn = [0,0,0,0,0,1,0,0,0,2,0,4,0,0,0,8,0,16,0,0,0,32,0,0,0,0,0,64,0,128]
  let range_size = end_num - start_num   # integers size for inputs range

  let kmax = (val - 2) div md + 1        # number of resgroups to input value
  var prms = newSeq[uint8](kmax)         # byte array of prime candidates init '0'
  let sqrtn = int(sqrt float64(val))     # compute integer sqrt of val
  var k = sqrtn div md                   # compute its resgroup value
  var (resk, r) = (sqrtn - md*k, 0)      # compute its residue value; set residue start posn
  while resk >= res[r]: r += 1           # find largest residue <= sqrtn posn in its resgroup
  let pcs_to_sqrtn = k*rescnt + r        # number of pcs <= sqrtn

  for i in 0..pcs_to_sqrtn:              # for r0..sqrtN primes mark their multiples
    let (k, r) = (i div rescnt, i mod rescnt) # update resgroup parameters
    if (prms[k] and uint8(1 shl r)) != 0: continue # skip pc if not prime
    let prm_r = res[r]                   # if prime save its residue value
    let prime = md*k + prm_r             # numerate its value
    let rem = start_num mod uint(prime)  # prime's modular distance to start_num
    if not(uint(prime) - rem <= range_size or rem == 0): continue # skip prime if no multiple in range
    for ri in res:                       # mark prime's multiples in prms
      let prod = prm_r * ri - 2          # compute cross-product for prm_r|ri pair
      let bit_r = uint8(bitn[prod mod md])     # bit mask value for prod's residue
      var kpm = k * (prime + ri) + prod div md # resgroup for prime's 1st multiple
      while kpm < kmax: prms[kpm] = prms[kpm] or bit_r; kpm += prime

  # 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)
  result = @[]                           # create empty dynamic array for primes
  for r, res_r in res:                   # along each restrack|row til end
    for k, resgroup in prms:             # for each resgroup along restrack
      if (resgroup and uint8(1 shl r)) == 0: # if bit location a prime
        let prime = md * k + res_r       # numerate its value, store if in range
        # check if prime has multiple in range, if so keep it, if not don't
        let rem = start_num mod prime.uint
        if (prime in res_0..val) and (prime.uint - rem <= range_size or rem == 0): result.add(prime)

 proc nextp_init(hi_r: int, kmin: uint, modpg: int, primes: seq[int], resinvrs: seq[int]): seq[uint64] =
  ## Initialize 'nextp' array for twinpair upper residue rhi in 'restwins'.
  ## Compute 1st prime multiple resgroups for each prime r0..sqrt(N) and
  ## store consecutively as lo_tp|hi_tp pairs for their restracks.
  var nextp = newSeq[uint64](primes.len * 2) # 1st mults array for twinpair
  let (r_hi, r_lo) = (hi_r, hi_r - 2)    # upper|lower twinpair residues vals
  for j, prime in primes:                # for each prime r0..sqrt(N)
    let k = (prime - 2) div modpg        # find the resgroup it's in
    let r = (prime - 2) mod modpg + 2    # and its residue value
    let r_inv = resinvrs[r]              # and its residue inverse
    var ri = (r_lo * r_inv - 2) mod modpg + 2 # compute r's ri for r_lo
    var ki = uint(k * (prime + ri) + (r * ri - 2) div modpg) # and 1st mult
    if ki < kmin:                        # if 1st mult index < start_num's
      ki = (kmin - ki) mod prime.uint    # how many indices short is it
      if ki > 0'u: ki = prime.uint - ki  # adjust index value into range
    else: ki = ki - kmin                 # else here, adjust index if it was >
    nextp[j * 2] = ki                    # lo_tp index val >= start of range
    ri = (r_hi * r_inv - 2) mod modpg + 2     # compute r's ri for r_hi
    ki = uint(k * (prime + ri) + (r * ri - 2) div modpg) # and 1st mult
    if ki < kmin:                        # if 1st mult index < start_num's
      ki = (kmin - ki) mod prime.uint    # how many indices short is it
      if ki > 0'u: ki = prime.uint - ki  # adjust index value into range
    else: ki = ki - kmin                 # else here, adjust index if it was >
    nextp[j * 2 or 1] = ki               # hi_tp index val >= start of range
  result = nextp

 proc twins_sieve(r_hi, kmin, kmax, Ks, start_num, end_num, modpg: uint, primes: seq[int], resinvrs: seq[int], indx: int) =
  ## Perform in thread the ssoz for given twinpair residues for Kmax resgroups.
  ## First create|init 'nextp' array of 1st prime mults for given twinpair,
  ## stored consequtively in 'nextp', and init seg array for Ks resgroups.
  ## For sieve, mark resgroup bits to '1' if either twinpair restrack is nonprime
  ## for primes mults resgroups, and update 'nextp' restrack slices acccordingly.
  ## Find last twin prime|sum for range, store at array[indx] for this twinpair.
  ## Can optionally compile to print mid twinprime values generated by twinpair.
  {.gcsafe.}:
   const S = 6                                # shift value for 64 bits
   const BMASK = (1 shl S) - 1                # bitmask val for 64 bits
   var (sum, Ki, Kn) = (0'u, kmin-1, Ks.int)  # init these parameters
   var (hi_tp, Kmax) = (0'u, kmax)            # max twinprime|resgroup val
   var seg = newSeq[uint](((Ks-1) shr S) + 1) # seg array for Ks resgroups
   if r_hi - 2 < (start_num - 2) mod modpg + 2: Ki.inc # ensure lo tps in range
   if r_hi > (end_num - 2) mod modpg + 2: Kmax.dec     # ensure hi tps in range
   var nextp = nextp_init(r_hi.int, Ki, modpg.int, primes, resinvrs) # init nextp array
   while Ki < Kmax:                      # for Ks resgroup size slices upto Kmax
     if Ks > (Kmax - Ki): Kn = int(Kmax - Ki) # adjust Kn size for last seg
     for j, prime in primes:             # for each prime r0..sqrt(N)
                                         # for lower twinpair residue track
       var k1 = nextp[j * 2].int         # starting from this resgroup in seg
       while k1 < Kn:                    # mark primenth resgroup bits prime mults
         seg[k1 shr S] = seg[k1 shr S] or (1'u shl (k1 and BMASK)).uint
         k1 += prime                     # set next prime multiple resgroup
       nextp[j * 2] = uint(k1 - Kn)      # save 1st resgroup in next eligible seg
                                         # for upper twinpair residue track
       var k2 = nextp[j * 2 or 1].int    # starting from this resgroup in seg
       while k2 < Kn:                    # mark primenth resgroup bits prime mults
         seg[k2 shr S] = seg[k2 shr S] or (1'u shl (k2 and BMASK)).uint
         k2 += prime                     # set next prime multiple resgroup
       nextp[j * 2 or 1] = uint(k2 - Kn) # save 1st resgroup in next eligible seg
                                         # set as nonprime unused bits in last seg[n]
                                         # so fast, do for every seg[i]
     seg[(Kn-1) shr S] = seg[(Kn-1) shr S] or (not 1'u shl ((Kn-1) and BMASK)).uint
     var cnt = 0                         # count the twinprimes in the segment
     for m in seg[0..(Kn-1) shr S]: cnt += (1 shl S) - countSetBits(m)
     if cnt > 0:                         # if segment has twin primes
       sum += cnt.uint                   # add segment count to total range count
       var upk = Kn - 1                  # from end of seg count back to largest tp
       while (seg[upk shr S] and (1'u shl (upk and BMASK)).uint) != 0: upk.dec
       hi_tp = Ki + upk.uint             # numerate its full resgroup value
     Ki += Ks                            # set 1st resgroup val of next seg slice
     if Ki < Kmax: (for b in 0..(Kn - 1) shr S: seg[b] = 0) # set seg to all primes
                                         # when sieve done for full range
                                         # numerate largest twinprime in segs
   hi_tp = if r_hi > end_num: 0'u else: hi_tp * modpg.uint + r_hi
   lastwins[indx] = if sum == 0: 1'u else: hi_tp # store final seg tp value
   cnts[indx] = sum                      # sum for twin pair

 proc twinprimes_ssoz() =
  ## Main routine to setup, time, and display results for twin primes sieve.
  let x = stdin.readline.split           # Inputs are 1 or 2 range values < 2**64
  var end_num = max(x[0].parseUInt, 3'u)
  var start_num = if x.len > 1: max(x[1].parseUInt, 3'u) else: 3'u
  if start_num > end_num: swap start_num, end_num
  start_num = start_num or 1             # if start_num even add 1
  end_num = (end_num - 1) or 1           # if end_num even subtract 1
  if end_num - start_num < 2: (start_num, end_num) = (7'u, 7'u)

  echo("threads = ", countProcessors())
  let ts = epochTime()                   # start timing sieve setup execution
                                         # select Pn, set sieving params for inputs
  let (modpg, res_0, Ks, Kmin, Kmax, Krange, 
      pairscnt, restwins, resinvrs) = setSieveParameters(start_num, end_num)

  # create sieve primes <= sqrt(end_num), only use those whose multiples within inputs range
  let primes: seq[int] = if end_num < 49: @[5]
      else: sozp5(int(sqrt float64(end_num)), res_0, start_num, end_num)

  echo("each of ", pairscnt, " threads has nextp[", 2, " x ", primes.len, "] array")

  var twinscnt = 0'u64                   # init twinprimes range count
  let lo_range = uint(restwins[0] - 3)   # lo_range = lo_tp - 1
  for tp in [3, 5, 11, 17]:              # excluded low tp values for PGs used
    if end_num == 3'u: break             # if 3 end of range, no twin primes
    if tp.uint in start_num..lo_range: twinscnt += 1 # cnt small tps if any

  let te = epochTime() - ts              # sieve setup time
  echo("setup time = ", te.formatFloat(ffDecimal, 6), " secs")
  echo("perform twinprimes ssoz sieve")
  let t1 = epochTime()                   # start timing ssoz sieve execution

  cnts = newSeq[uint](pairscnt)          # array to hold count of tps for each thread
  lastwins = newSeq[uint](pairscnt)      # array to hold largest tp for each thread

  #parallel:                             # perform in parallel
  for indx, r_hi in restwins:            # for each twin pair row index
    spawn twins_sieve(r_hi.uint, Kmin, Kmax, Ks, start_num, end_num, modpg, primes, resinvrs, indx)
    stdout.write("\r", (indx + 1), " of ", pairscnt, " twinpairs done")
  sync()                                 # when all the threads finish

  var last_twin = 0'u                    # find largest twin prime in range
  for i in 0 ..< pairscnt:               # and determine total twin primes count
    twinscnt += cnts[i]
    if last_twin < lastwins[i]: last_twin = lastwins[i]
  if end_num == 5'u and twinscnt == 1: last_twin = 5'u
  var Kn = Krange mod Ks                 # set number of resgroups in last slice
  if Kn == 0: Kn = Ks                    # if multiple of seg size set to seg size
  let t2 = epochTime() - t1              # sieve execution time

  echo("\nsieve time = ", t2.formatFloat(ffDecimal, 3), " secs")
  echo("total time = ", (t2 + te).formatFloat(ffDecimal, 3), " secs")
  echo("last segment = ", Kn, " resgroups; segment slices = ", (Krange - 1) div Ks + 1)
  echo("total twins = ", twinscnt, "; last twin = ", last_twin - 1, "+/-1")

 twinprimes_ssoz()
	#[
	This Nim source file is a multiple threaded implementation to perform an
	extremely fast Segmented Sieve of Zakiya (SSoZ) to find Twin Primes <= N.

	Inputs are single values N, or ranges N1 and N2, of 64-bits, 0 -- 2^64 - 1.
	Output is the number of twin primes <= N, or in range N1 to N2; the last
	twin 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
	would be needed to optimize for other hardware systems (ARM, PowerPC, etc).

	For Nim >= 2.0.0 compile as:
	$ nim c --cc:gcc --mm:arc --d:danger --d:release --threads:on --d:useMalloc twinprimes_ssozgist_new.nim

	Then run executable: $ ./twinprimes_ssozgist1 <cr>, and enter range values.
	Or alternatively: $ echo <range1> <range2> \| ./twinprimes_ssoz
	Range values can be provided in either order (larger or smaller first).

	This source file, and updates, will be available here:
	https://gist.github.com/jzakiya/6c7e1868bd749a6b1add62e3e3b2341e

	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 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
	Version Date: 2024/08/12
	]#

	import math # for sqrt, gcd, mod functions
	import strutils, typetraits # for number input
	import times, os # for timing code execution
	import osproc # for getting threads count
	import threadpool # for parallel processing
	import algorithm # for sort
	import bitops # for countSetBits
	{.experimental: "parallel".} # required to use 'parallel' (>= 1.4.x)

	proc modinv(a0, b0: int): int =
	## Compute modular inverse a^-1 of a to base b, e.g. a*(a^-1) mod b = 1.
	var (a, b, x0) = (a0, b0, 0)
	result = 1
	if b == 1: return
	while a > 1:
	result -= (a div b) * x0
	a = a mod b
	swap a, b
	swap x0, result
	if result < 0: result += b0

	proc genPGparameters(prime: int): (int, int, int, seq[int], seq[int]) =
	## Create prime generator parameters for given Pn
	echo("using Prime Generator parameters for P", prime)
	let primes = [2, 3, 5, 7, 11, 13, 17, 19, 23]
	var (modpg, res_0) = (1, 0) # compute Pn's modulus and res_0 value
	for prm in primes: (res_0 = prm; if prm > prime: break; modpg *= prm)

	var restwins: seq[int] = @[] # save upper twin pair residues here
	var inverses = newSeq[int](modpg+2) # save Pn's residues inverses here
	var (rc, inc, res) = (5, 2, 0) # use P3's PGS to generate residue candidates
	while rc < (modpg shr 1): # find PG's 1st half residues
	if gcd(modpg, rc) == 1: # if rc is a residue
	let mc = modpg - rc # create its modular complement
	inverses[rc] = modinv(rc,modpg) # save rc amd mc inverses
	inverses[mc] = modinv(mc,modpg) # if in twinpair save both hi residues
	if res + 2 == rc: restwins.add(rc); restwins.add(mc + 2)
	res = rc # save current found residue
	rc += inc; inc = inc xor 0b110 # create next P3 sequence rc: 5 7 11 13 17 19 ...

	restwins.sort(system.cmp[int]); restwins.add(modpg + 1) # last residue is last hi_tp
	inverses[modpg + 1] = 1; inverses[modpg - 1] = modpg - 1 # last 2 resdiues are self inverses
	(modpg, res_0, restwins.len, restwins, inverses)

	# Global parameters
	var
	cnts: seq[uint] # holds twinprimes count for each twinpair
	lastwins: seq[uint] # holds largest twinprime for each twinpair

	proc setSieveParameters(start_num, end_num: uint): (uint, int, uint, uint, uint, uint, int, seq[int], seq[int]) =
	## Select at runtime best PG and segment size factor to use for input value.
	## These are good estimates derived from PG data profiling. Can be improved.
	let range = end_num - start_num
	var (Bn, pg) = (0, 3)
	if end_num < 49'u:
	Bn = 1; pg = 3
	elif range < 77_000_000:
	Bn = 16; pg = 5
	elif range < 1_100_000_000'u:
	Bn = 32; pg = 7
	elif range < 35_500_000_000'u:
	Bn = 64; pg = 11
	elif range < 14_000_000_000_000'u:
	pg = 13
	if range > 7_000_000_000_000'u: Bn = 384
	elif range > 2_500_000_000_000'u: Bn = 320
	elif range > 250_000_000_000'u: Bn = 196
	else: Bn = 128
	else:
	Bn = 384; pg = 17
	let (modpg, res_0, pairscnt, restwins, resinvrs) = genPGparameters(pg)
	let Kmin = (start_num - 2) div modpg.uint + 1 # number of resgroups to start_num
	let Kmax = (end_num - 2) div modpg.uint + 1 # number of resgroups to end_num
	let krange = Kmax - Kmin + 1 # number of range resgroups, at least 1
	let n = if krange < 37_500_000_000_000'u: 10 elif krange < 975_000_000_000_000'u: 16 else: 20
	let B = uint(Bn * 1024 * n) # set seg size to optimize for selected PG
	let Ks = if krange < B: krange else: B # segments resgroups size

	echo("segment size = ",Ks," resgroups; seg array is [1 x ",((Ks-1) shr 6) + 1,"] 64-bits")
	let maxpairs = krange.int * pairscnt; # maximum number of twinprime pcs
	echo("twinprime candidates = ", maxpairs, "; resgroups = ", krange);
	(modpg.uint, res_0, Ks, Kmin, Kmax, krange, pairscnt, restwins, resinvrs)

	proc sozp5(val: int \| int64, res_0: int, start_num, end_num: uint): seq[int] =
	## Return the primes r0..sqrt(end_num) within range (start_num...end_num)
	let (md, rescnt) = (30, 8) # P5's modulus and residues count
	let res = [7,11,13,17,19,23,29,31] # P5's residues list
	let bitn = [0,0,0,0,0,1,0,0,0,2,0,4,0,0,0,8,0,16,0,0,0,32,0,0,0,0,0,64,0,128]
	let range_size = end_num - start_num # integers size for inputs range

	let kmax = (val - 2) div md + 1 # number of resgroups to input value
	var prms = newSeq[uint8](kmax) # byte array of prime candidates init '0'
	let sqrtn = int(sqrt float64(val)) # compute integer sqrt of val
	var k = sqrtn div md # compute its resgroup value
	var (resk, r) = (sqrtn - md*k, 0) # compute its residue value; set residue start posn
	while resk >= res[r]: r += 1 # find largest residue <= sqrtn posn in its resgroup
	let pcs_to_sqrtn = k*rescnt + r # number of pcs <= sqrtn

	for i in 0..pcs_to_sqrtn: # for r0..sqrtN primes mark their multiples
	let (k, r) = (i div rescnt, i mod rescnt) # update resgroup parameters
	if (prms[k] and uint8(1 shl r)) != 0: continue # skip pc if not prime
	let prm_r = res[r] # if prime save its residue value
	let prime = md*k + prm_r # numerate its value
	let rem = start_num mod uint(prime) # prime's modular distance to start_num
	if not(uint(prime) - rem <= range_size or rem == 0): continue # skip prime if no multiple in range
	for ri in res: # mark prime's multiples in prms
	let prod = prm_r * ri - 2 # compute cross-product for prm_r\|ri pair
	let bit_r = uint8(bitn[prod mod md]) # bit mask value for prod's residue
	var kpm = k * (prime + ri) + prod div md # resgroup for prime's 1st multiple
	while kpm < kmax: prms[kpm] = prms[kpm] or bit_r; kpm += prime

	# 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)
	result = @[] # create empty dynamic array for primes
	for r, res_r in res: # along each restrack\|row til end
	for k, resgroup in prms: # for each resgroup along restrack
	if (resgroup and uint8(1 shl r)) == 0: # if bit location a prime
	let prime = md * k + res_r # numerate its value, store if in range
	# check if prime has multiple in range, if so keep it, if not don't
	let rem = start_num mod prime.uint
	if (prime in res_0..val) and (prime.uint - rem <= range_size or rem == 0): result.add(prime)

	proc nextp_init(hi_r: int, kmin: uint, modpg: int, primes: seq[int], resinvrs: seq[int]): seq[uint64] =
	## Initialize 'nextp' array for twinpair upper residue rhi in 'restwins'.
	## Compute 1st prime multiple resgroups for each prime r0..sqrt(N) and
	## store consecutively as lo_tp\|hi_tp pairs for their restracks.
	var nextp = newSeq[uint64](primes.len * 2) # 1st mults array for twinpair
	let (r_hi, r_lo) = (hi_r, hi_r - 2) # upper\|lower twinpair residues vals
	for j, prime in primes: # for each prime r0..sqrt(N)
	let k = (prime - 2) div modpg # find the resgroup it's in
	let r = (prime - 2) mod modpg + 2 # and its residue value
	let r_inv = resinvrs[r] # and its residue inverse
	var ri = (r_lo * r_inv - 2) mod modpg + 2 # compute r's ri for r_lo
	var ki = uint(k * (prime + ri) + (r * ri - 2) div modpg) # and 1st mult
	if ki < kmin: # if 1st mult index < start_num's
	ki = (kmin - ki) mod prime.uint # how many indices short is it
	if ki > 0'u: ki = prime.uint - ki # adjust index value into range
	else: ki = ki - kmin # else here, adjust index if it was >
	nextp[j * 2] = ki # lo_tp index val >= start of range
	ri = (r_hi * r_inv - 2) mod modpg + 2 # compute r's ri for r_hi
	ki = uint(k * (prime + ri) + (r * ri - 2) div modpg) # and 1st mult
	if ki < kmin: # if 1st mult index < start_num's
	ki = (kmin - ki) mod prime.uint # how many indices short is it
	if ki > 0'u: ki = prime.uint - ki # adjust index value into range
	else: ki = ki - kmin # else here, adjust index if it was >
	nextp[j * 2 or 1] = ki # hi_tp index val >= start of range
	result = nextp

	proc twins_sieve(r_hi, kmin, kmax, Ks, start_num, end_num, modpg: uint, primes: seq[int], resinvrs: seq[int], indx: int) =
	## Perform in thread the ssoz for given twinpair residues for Kmax resgroups.
	## First create\|init 'nextp' array of 1st prime mults for given twinpair,
	## stored consequtively in 'nextp', and init seg array for Ks resgroups.
	## For sieve, mark resgroup bits to '1' if either twinpair restrack is nonprime
	## for primes mults resgroups, and update 'nextp' restrack slices acccordingly.
	## Find last twin prime\|sum for range, store at array[indx] for this twinpair.
	## Can optionally compile to print mid twinprime values generated by twinpair.
	{.gcsafe.}:
	const S = 6 # shift value for 64 bits
	const BMASK = (1 shl S) - 1 # bitmask val for 64 bits
	var (sum, Ki, Kn) = (0'u, kmin-1, Ks.int) # init these parameters
	var (hi_tp, Kmax) = (0'u, kmax) # max twinprime\|resgroup val
	var seg = newSeq[uint](((Ks-1) shr S) + 1) # seg array for Ks resgroups
	if r_hi - 2 < (start_num - 2) mod modpg + 2: Ki.inc # ensure lo tps in range
	if r_hi > (end_num - 2) mod modpg + 2: Kmax.dec # ensure hi tps in range
	var nextp = nextp_init(r_hi.int, Ki, modpg.int, primes, resinvrs) # init nextp array
	while Ki < Kmax: # for Ks resgroup size slices upto Kmax
	if Ks > (Kmax - Ki): Kn = int(Kmax - Ki) # adjust Kn size for last seg
	for j, prime in primes: # for each prime r0..sqrt(N)
	# for lower twinpair residue track
	var k1 = nextp[j * 2].int # starting from this resgroup in seg
	while k1 < Kn: # mark primenth resgroup bits prime mults
	seg[k1 shr S] = seg[k1 shr S] or (1'u shl (k1 and BMASK)).uint
	k1 += prime # set next prime multiple resgroup
	nextp[j * 2] = uint(k1 - Kn) # save 1st resgroup in next eligible seg
	# for upper twinpair residue track
	var k2 = nextp[j * 2 or 1].int # starting from this resgroup in seg
	while k2 < Kn: # mark primenth resgroup bits prime mults
	seg[k2 shr S] = seg[k2 shr S] or (1'u shl (k2 and BMASK)).uint
	k2 += prime # set next prime multiple resgroup
	nextp[j * 2 or 1] = uint(k2 - Kn) # save 1st resgroup in next eligible seg
	# set as nonprime unused bits in last seg[n]
	# so fast, do for every seg[i]
	seg[(Kn-1) shr S] = seg[(Kn-1) shr S] or (not 1'u shl ((Kn-1) and BMASK)).uint
	var cnt = 0 # count the twinprimes in the segment
	for m in seg[0..(Kn-1) shr S]: cnt += (1 shl S) - countSetBits(m)
	if cnt > 0: # if segment has twin primes
	sum += cnt.uint # add segment count to total range count
	var upk = Kn - 1 # from end of seg count back to largest tp
	while (seg[upk shr S] and (1'u shl (upk and BMASK)).uint) != 0: upk.dec
	hi_tp = Ki + upk.uint # numerate its full resgroup value
	Ki += Ks # set 1st resgroup val of next seg slice
	if Ki < Kmax: (for b in 0..(Kn - 1) shr S: seg[b] = 0) # set seg to all primes
	# when sieve done for full range
	# numerate largest twinprime in segs
	hi_tp = if r_hi > end_num: 0'u else: hi_tp * modpg.uint + r_hi
	lastwins[indx] = if sum == 0: 1'u else: hi_tp # store final seg tp value
	cnts[indx] = sum # sum for twin pair

	proc twinprimes_ssoz() =
	## Main routine to setup, time, and display results for twin primes sieve.
	let x = stdin.readline.split # Inputs are 1 or 2 range values < 2**64
	var end_num = max(x[0].parseUInt, 3'u)
	var start_num = if x.len > 1: max(x[1].parseUInt, 3'u) else: 3'u
	if start_num > end_num: swap start_num, end_num
	start_num = start_num or 1 # if start_num even add 1
	end_num = (end_num - 1) or 1 # if end_num even subtract 1
	if end_num - start_num < 2: (start_num, end_num) = (7'u, 7'u)

	echo("threads = ", countProcessors())
	let ts = epochTime() # start timing sieve setup execution
	# select Pn, set sieving params for inputs
	let (modpg, res_0, Ks, Kmin, Kmax, Krange,
	pairscnt, restwins, resinvrs) = setSieveParameters(start_num, end_num)

	# create sieve primes <= sqrt(end_num), only use those whose multiples within inputs range
	let primes: seq[int] = if end_num < 49: @[5]
	else: sozp5(int(sqrt float64(end_num)), res_0, start_num, end_num)

	echo("each of ", pairscnt, " threads has nextp[", 2, " x ", primes.len, "] array")

	var twinscnt = 0'u64 # init twinprimes range count
	let lo_range = uint(restwins[0] - 3) # lo_range = lo_tp - 1
	for tp in [3, 5, 11, 17]: # excluded low tp values for PGs used
	if end_num == 3'u: break # if 3 end of range, no twin primes
	if tp.uint in start_num..lo_range: twinscnt += 1 # cnt small tps if any

	let te = epochTime() - ts # sieve setup time
	echo("setup time = ", te.formatFloat(ffDecimal, 6), " secs")
	echo("perform twinprimes ssoz sieve")
	let t1 = epochTime() # start timing ssoz sieve execution

	cnts = newSeq[uint](pairscnt) # array to hold count of tps for each thread
	lastwins = newSeq[uint](pairscnt) # array to hold largest tp for each thread

	#parallel: # perform in parallel
	for indx, r_hi in restwins: # for each twin pair row index
	spawn twins_sieve(r_hi.uint, Kmin, Kmax, Ks, start_num, end_num, modpg, primes, resinvrs, indx)
	stdout.write("\r", (indx + 1), " of ", pairscnt, " twinpairs done")
	sync() # when all the threads finish

	var last_twin = 0'u # find largest twin prime in range
	for i in 0 ..< pairscnt: # and determine total twin primes count
	twinscnt += cnts[i]
	if last_twin < lastwins[i]: last_twin = lastwins[i]
	if end_num == 5'u and twinscnt == 1: last_twin = 5'u
	var Kn = Krange mod Ks # set number of resgroups in last slice
	if Kn == 0: Kn = Ks # if multiple of seg size set to seg size
	let t2 = epochTime() - t1 # sieve execution time

	echo("\nsieve time = ", t2.formatFloat(ffDecimal, 3), " secs")
	echo("total time = ", (t2 + te).formatFloat(ffDecimal, 3), " secs")
	echo("last segment = ", Kn, " resgroups; segment slices = ", (Krange - 1) div Ks + 1)
	echo("total twins = ", twinscnt, "; last twin = ", last_twin - 1, "+/-1")

	twinprimes_ssoz()