Twin primes: Difference between revisions

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Twin primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

Twin primes are pairs of natural numbers   (P1  and  P2)   that satisfy the following:

  1.     P1   and   P2   are primes
  2.     P1  +  2   =   P2


Task

Write a program that displays the number of pairs of twin primes that can be found under a user-specified number
(P1 < user-specified number & P2 < user-specified number).


Extension
  1. Find all twin prime pairs under 100000, 10000000 and 1000000000.
  2. What is the time complexity of the program? Are there ways to reduce computation time?


Examples
> Search Size: 100
> 8 twin prime pairs.

> Search Size: 1000
> 35 twin prime pairs.


Also see



ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32

Simplifies array bound checking by using the equivalent definition of twin primes: p and p - 2. <lang algol68>BEGIN 

   # count twin primes (where p and p - 2 are prime)                             #
   PR heap=128M PR # set heap memory size for Algol 68G                          #
   # sieve of Eratosthenes: sets s[i] to TRUE if i is a prime, FALSE otherwise   #
   PROC sieve = ( REF[]BOOL s )VOID:
        BEGIN
           FOR i TO UPB s DO s[ i ] := TRUE OD;
           s[ 1 ] := FALSE;
           FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO
               IF s[ i ] THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := FALSE OD FI
           OD
        END # sieve # ;
   # find the maximum number to search for twin primes                           #
   INT max;
   print( ( "Maximum: " ) );
   read( ( max, newline ) );
   INT max number = max;
   # construct a sieve of primes up to the maximum number                        #
   [ 1 : max number ]BOOL primes;
   sieve( primes );
   # count the twin primes                                                       #
   # note 2 cannot be one of the primes in a twin prime pair, so we start at 3   #
   INT twin count := 0;
   FOR p FROM 3 BY 2 TO max number - 1 DO IF primes[ p ] AND primes[ p - 2 ] THEN twin count +:= 1 FI OD;
   print( ( "twin prime pairs below  ", whole( max number, 0 ), ": ", whole( twin count, 0 ), newline ) )

END</lang>

Output:
Maximum: 10
twin prime pairs below  10: 2

Maximum: 100
twin prime pairs below  100: 8

Maximum: 1000
twin prime pairs below  1000: 35

Maximum: 10000
twin prime pairs below  10000: 205

Maximum: 100000
twin prime pairs below  100000: 1224

Maximum: 1000000
twin prime pairs below  1000000: 8169

Maximum: 10000000
twin prime pairs below  10000000: 58980

C

<lang c>#include <stdbool.h>

  1. include <stdint.h>
  2. include <stdio.h>

bool isPrime(int64_t n) { 

   int64_t i;

   if (n < 2)       return false;
   if (n % 2 == 0)  return n == 2;
   if (n % 3 == 0)  return n == 3;
   if (n % 5 == 0)  return n == 5;
   if (n % 7 == 0)  return n == 7;
   if (n % 11 == 0) return n == 11;
   if (n % 13 == 0) return n == 13;
   if (n % 17 == 0) return n == 17;
   if (n % 19 == 0) return n == 19;

   for (i = 23; i * i <= n; i += 2) {
       if (n % i == 0) return false;
   }

   return true;

}

int countTwinPrimes(int limit) { 

   int count = 0;

   //       2          3          4
   int64_t p3 = true, p2 = true, p1 = false;
   int64_t i;

   for (i = 5; i <= limit; i++) {
       p3 = p2;
       p2 = p1;
       p1 = isPrime(i);
       if (p3 && p1) {
           count++;
       }
   }
   return count;

}

void test(int limit) { 

   int count = countTwinPrimes(limit);
   printf("Number of twin prime pairs less than %d is %d\n", limit, count);

}

int main() { 

   test(10);
   test(100);
   test(1000);
   test(10000);
   test(100000);
   test(1000000);
   test(10000000);
   test(100000000);
   return 0;

}</lang>

Output:
Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1000 is 35
Number of twin prime pairs less than 10000 is 205
Number of twin prime pairs less than 100000 is 1224
Number of twin prime pairs less than 1000000 is 8169
Number of twin prime pairs less than 10000000 is 58980
Number of twin prime pairs less than 100000000 is 440312

C++

Library: Primesieve

The primesieve library includes the functionality required for this task. RC already has plenty of C++ examples showing how to generate prime numbers from scratch. (The module Math::Primesieve, which is used by the Raku example on this page, is implemented on top of this library.) <lang cpp>#include <cstdint>

  1. include <iostream>
  2. include <string>
  3. include <primesieve.hpp>

void print_twin_prime_count(long long limit) { 

   std::cout << "Number of twin prime pairs less than " << limit
       << " is " << (limit > 0 ? primesieve::count_twins(0, limit - 1) : 0) << '\n';

}

int main(int argc, char** argv) { 

   std::cout.imbue(std::locale(""));
   if (argc > 1) {
       // print number of twin prime pairs less than limits specified
       // on the command line
       for (int i = 1; i < argc; ++i) {
           try {
               print_twin_prime_count(std::stoll(argv[i]));
           } catch (const std::exception& ex) {
               std::cerr << "Cannot parse limit from '" << argv[i] << "'\n";
           }
       }
   } else {
       // if no limit was specified then show the number of twin prime
       // pairs less than powers of 10 up to 100 billion
       uint64_t limit = 10;
       for (int power = 1; power < 12; ++power, limit *= 10)
           print_twin_prime_count(limit);
   }
   return 0;

}</lang>

Output:
Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1,000 is 35
Number of twin prime pairs less than 10,000 is 205
Number of twin prime pairs less than 100,000 is 1,224
Number of twin prime pairs less than 1,000,000 is 8,169
Number of twin prime pairs less than 10,000,000 is 58,980
Number of twin prime pairs less than 100,000,000 is 440,312
Number of twin prime pairs less than 1,000,000,000 is 3,424,506
Number of twin prime pairs less than 10,000,000,000 is 27,412,679
Number of twin prime pairs less than 100,000,000,000 is 224,376,048

Delphi

Library: System.SysUtils
Translation of: Wren

<lang Delphi> program Primes;

{$APPTYPE CONSOLE}

{$R *.res}

uses 

 System.SysUtils;

function IsPrime(a: UInt64): Boolean; var 

 d: UInt64;

begin 

 if (a < 2) then
   exit(False);

 if (a mod 2) = 0 then
   exit(a = 2);

 if (a mod 3) = 0 then
   exit(a = 3);

 d := 5;

 while (d * d <= a) do
 begin
   if (a mod d = 0) then
     Exit(false);
   inc(d, 2);

   if (a mod d = 0) then
     Exit(false);
   inc(d, 4);
 end;

 Result := True;

end;


function Sieve(limit: UInt64): TArray<Boolean>; var 

 p, p2, i: UInt64;

begin 

 inc(limit);
 SetLength(Result, limit);
 FillChar(Result[2], sizeof(Boolean) * limit - 2, 0); // all false except 1,2
 FillChar(Result[0], sizeof(Boolean) * 2, 1); // 1,2 are true

 p := 3;
 while true do
 begin
   p2 := p * p;
   if p2 >= limit then
     break;

   i := p2;
   while i < limit do
   begin
     Result[i] := true;
     inc(i, 2 * p);
   end;

   while true do
   begin
     inc(p, 2);
     if not Result[p] then
       Break;
   end;
 end;

end;

function Commatize(const n: UInt64): string; var 

 str: string;
 digits: Integer;
 i: Integer;

begin 

 Result := ;
 str := n.ToString;
 digits := str.Length;

 for i := 1 to digits do
 begin
   if ((i > 1) and (((i - 1) mod 3) = (digits mod 3))) then
     Result := Result + ',';
   Result := Result + str[i];
 end;

end;

var 

 limit, start, twins: UInt64;
 c: TArray<Boolean>;
 i, j: UInt64;

begin 

 c := Sieve(Trunc(1e9 - 1));
 limit := 10;
 start := 3;
 twins := 0;
 for i := 1 to 9 do
 begin
   j := start;
   while j < limit do
   begin
     if (not c[j]) and (not c[j - 2]) then
       inc(twins);
     inc(j, 2);
   end;
   Writeln(Format('Under %14s there are %10s pairs of twin primes.', [commatize
     (limit), commatize(twins)]));

   start := limit + 1;
   limit := 10 * limit;
 end;

 readln;

end.

</lang>

Output:
Under             10 there are          2 pairs of twin primes.
Under            100 there are          8 pairs of twin primes.
Under          1,000 there are         35 pairs of twin primes.
Under         10,000 there are        205 pairs of twin primes.
Under        100,000 there are      1,224 pairs of twin primes.
Under      1,000,000 there are      8,169 pairs of twin primes.
Under     10,000,000 there are     58,980 pairs of twin primes.
Under    100,000,000 there are    440,312 pairs of twin primes.
Under  1,000,000,000 there are  3,424,506 pairs of twin primes.

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> printfn "twin primes below 100000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 1000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=1000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 10000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=10000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 100000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 1000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=1000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 10000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=10000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) printfn "twin primes below 100000000000: %d" (primes64()|>Seq.takeWhile(fun n->n<=100000000000L)|>Seq.pairwise|>Seq.filter(fun(n,g)->g=n+2L)|>Seq.length) </lang>

Output:
twin primes below 100000: 1224
Real: 00:00:00.003, CPU: 00:00:00.015, GC gen0: 0, gen1: 0, gen2: 0

twin primes below 1000000: 8169
Real: 00:00:00.021, CPU: 00:00:00.031, GC gen0: 3, gen1: 3, gen2: 0

twin primes below 10000000: 58980
Real: 00:00:00.154, CPU: 00:00:00.171, GC gen0: 19, gen1: 19, gen2: 0

twin primes below 100000000: 440312
Real: 00:00:01.400, CPU: 00:00:01.406, GC gen0: 162, gen1: 162, gen2: 0

twin primes below 1000000000: 3424506
Real: 00:00:12.682, CPU: 00:00:12.671, GC gen0: 1428, gen1: 1426, gen2: 1

twin primes below 10000000000: 27412679
Real: 00:02:04.441, CPU: 00:02:04.406, GC gen0: 12771, gen1: 12768, gen2: 2

twin primes below 100000000000: 224376048
Real: 00:23:00.853, CPU: 00:23:00.125, GC gen0: 115562, gen1: 115536, gen2: 14

Factor

Works with: Factor version 0.99 2020-07-03

<lang factor>USING: io kernel math math.parser math.primes.erato math.ranges sequences tools.memory.private ;

twin-pair-count ( n -- count )
   [ 5 swap 2 <range> ] [ sieve ] bi
   [ over 2 - over [ marked-prime? ] 2bi@ and ] curry count ;

"Search size: " write flush readln string>number twin-pair-count commas write " twin prime pairs." print</lang>

Output:
Search size: 100,000
1,224 twin prime pairs.

Search size: 10,000,000
58,980 twin prime pairs.

Search size: 1,000,000,000
3,424,506 twin prime pairs.

Go

Translation of: Wren

<lang go>package main

import "fmt"

func sieve(limit uint64) []bool { 

   limit++
   // True denotes composite, false denotes prime.
   c := make([]bool, limit) // all false by default
   c[0] = true
   c[1] = true
   // no need to bother with even numbers over 2 for this task
   p := uint64(3) // Start from 3.
   for {
       p2 := p * p
       if p2 >= limit {
           break
       }
       for i := p2; i < limit; i += 2 * p {
           c[i] = true
       }
       for {
           p += 2
           if !c[p] {
               break
           }
       }
   }
   return c

}

func commatize(n int) string { 

   s := fmt.Sprintf("%d", n)
   if n < 0 {
       s = s[1:]
   }
   le := len(s)
   for i := le - 3; i >= 1; i -= 3 {
       s = s[0:i] + "," + s[i:]
   }
   if n >= 0 {
       return s
   }
   return "-" + s

}

func main() { 

   c := sieve(1e10 - 1)
   limit := 10
   start := 3
   twins := 0
   for i := 1; i < 11; i++ {
       for i := start; i < limit; i += 2 {
           if !c[i] && !c[i-2] {
               twins++
           }
       }
       fmt.Printf("Under %14s there are %10s pairs of twin primes.\n", commatize(limit), commatize(twins))
       start = limit + 1
       limit *= 10
   }

}</lang>

Output:
Under             10 there are          2 pairs of twin primes.
Under            100 there are          8 pairs of twin primes.
Under          1,000 there are         35 pairs of twin primes.
Under         10,000 there are        205 pairs of twin primes.
Under        100,000 there are      1,224 pairs of twin primes.
Under      1,000,000 there are      8,169 pairs of twin primes.
Under     10,000,000 there are     58,980 pairs of twin primes.
Under    100,000,000 there are    440,312 pairs of twin primes.
Under  1,000,000,000 there are  3,424,506 pairs of twin primes.
Under 10,000,000,000 there are 27,412,679 pairs of twin primes.

Alternative using primegen package

See Extensible prime generator#Go. <lang go>package main

import ( 

   "fmt"
   "github.com/jbarham/primegen.go"

)

func main() { 

   p := primegen.New()
   count := 0
   previous := uint64(0)
   power := 1
   limit := uint64(10)
   for {
       prime := p.Next()
       if prime >= limit {
           fmt.Printf("Number of twin prime pairs less than %d: %d\n", limit, count)
           power++
           if power > 10 {
               break
           }
           limit *= 10
       }
       if previous > 0 && prime == previous + 2 {
           count++
       }
       previous = prime
   }

}</lang>

Output:
Number of twin prime pairs less than 10: 2
Number of twin prime pairs less than 100: 8
Number of twin prime pairs less than 1000: 35
Number of twin prime pairs less than 10000: 205
Number of twin prime pairs less than 100000: 1224
Number of twin prime pairs less than 1000000: 8169
Number of twin prime pairs less than 10000000: 58980
Number of twin prime pairs less than 100000000: 440312
Number of twin prime pairs less than 1000000000: 3424506
Number of twin prime pairs less than 10000000000: 27412679

Java

BigInteger Implementation: <lang Java> import java.math.BigInteger; import java.util.Scanner;

public class twinPrimes { 

   public static void main(String[] args) {
       Scanner input = new Scanner(System.in);
       System.out.println("Search Size: ");
       BigInteger max = input.nextBigInteger();
       int counter = 0;
       for(BigInteger x = new BigInteger("3"); x.compareTo(max) <= 0; x = x.add(BigInteger.ONE)){
           BigInteger sqrtNum = x.sqrt().add(BigInteger.ONE);
           if(x.add(BigInteger.TWO).compareTo(max) <= 0) {
               counter += findPrime(x.add(BigInteger.TWO), x.add(BigInteger.TWO).sqrt().add(BigInteger.ONE)) && findPrime(x, sqrtNum) ? 1 : 0;
           }
       }
       System.out.println(counter + " twin prime pairs.");
   }
   public static boolean findPrime(BigInteger x, BigInteger sqrtNum){
       for(BigInteger divisor = BigInteger.TWO; divisor.compareTo(sqrtNum) <= 0; divisor = divisor.add(BigInteger.ONE)){
           if(x.remainder(divisor).compareTo(BigInteger.ZERO) == 0){
               return false;
           }
       }
       return true;
   }

} </lang>

Output:
> Search Size: 
> 100
> 8 twin prime pairs.

> Search Size: 
> 1000
> 35 twin prime pairs.


Julia

<lang julia>using Formatting, Primes

function counttwinprimepairsbetween(n1, n2) 

   npairs, t = 0, nextprime(n1)
   while t < n2
       p = nextprime(t + 1)
       if p - t == 2
           npairs += 1
       end
       t = p
   end
   return npairs

end

for t2 in (10).^collect(2:8) 

   paircount = counttwinprimepairsbetween(1, t2)
   println("Under", lpad(format(t2, commas=true), 12), " there are",
       lpad(format(paircount, commas=true), 8), " pairs of twin primes.")

end

</lang>

Output:
Under         100 there are       8 pairs of twin primes.
Under       1,000 there are      35 pairs of twin primes.
Under      10,000 there are     205 pairs of twin primes.
Under     100,000 there are   1,224 pairs of twin primes.
Under   1,000,000 there are   8,169 pairs of twin primes.
Under  10,000,000 there are  58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.

Extension to large n and other tuples

Task Extension: to get primes up to a billion, it becomes important to cache the results so that large numbers do not need to be factored more than once. This trades memory for speed. The time complexity is dominated by the prime sieve time used to create the primes mask, which is n log(log n).

We can generalize pairs to reflect any tuple of integer differences between the first prime and the successive primes: see http://www.rosettacode.org/wiki/Successive_prime_differences.

If we ignore the first difference from the index prime with itself (always 0), we can express a prime pair as a difference tuple of (2,), and a prime quadruplet such as [11, 13, 17, 19] as the tuple starting with 11 of type (2, 6, 8). <lang julia>using Formatting, Primes

const PMAX = 1_000_000_000 const pmb = primesmask(PMAX) const primestoabillion = [i for i in 2:PMAX if pmb[i]]

tuplefitsat(k, tup, arr) = all(i -> arr[k + i] - arr[k] == tup[i], 1:length(tup))

function countprimetuples(tup, n) 

   arr =  filter(i -> i <= n, primestoabillion)
   return count(k -> tuplefitsat(k, tup, arr), 1:length(arr) - length(tup))

end

println("Count of prime pairs from 1 to 1 billion: ", 

   format(countprimetuples((2,), 1000000000), commas=true))

println("Count of a form of prime quads from 1 to 1 million: ", 

   format(countprimetuples((2, 6, 8), 1000000), commas=true))

println("Count of a form of prime octets from 1 to 1 million: ", 

   format(countprimetuples((2, 6, 12, 14, 20, 24, 26), 1000000), commas=true))

</lang>

Output:
Count of prime pairs from 1 to 1 billion: 3,424,506
Count of a form of prime quads from 1 to 1 million: 166
Count of a form of prime octets from 1 to 1 million: 3

Kotlin

Translation of: Java

<lang scala>import java.math.BigInteger import java.util.*

fun main() { 

   val input = Scanner(System.`in`)
   println("Search Size: ")
   val max = input.nextBigInteger()
   var counter = 0
   var x = BigInteger("3")
   while (x <= max) {
       val sqrtNum = x.sqrt().add(BigInteger.ONE)
       if (x.add(BigInteger.TWO) <= max) {
           counter += if (findPrime(
                   x.add(BigInteger.TWO),
                   x.add(BigInteger.TWO).sqrt().add(BigInteger.ONE)
               ) && findPrime(x, sqrtNum)
           ) 1 else 0
       }
       x = x.add(BigInteger.ONE)
   }
   println("$counter twin prime pairs.")

}

fun findPrime(x: BigInteger, sqrtNum: BigInteger?): Boolean { 

   var divisor = BigInteger.TWO
   while (divisor <= sqrtNum) {
       if (x.remainder(divisor).compareTo(BigInteger.ZERO) == 0) {
           return false
       }
       divisor = divisor.add(BigInteger.ONE)
   }
   return true

}</lang>

Perl

<lang perl>use strict; use warnings;

use Primesieve;

sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }

printf "Twin prime pairs less than %14s: %s\n", comma(10**$_), comma count_twins(1, 10**$_) for 1..10;</lang>

Output:
Twin prime pairs less than             10: 2
Twin prime pairs less than            100: 8
Twin prime pairs less than          1,000: 35
Twin prime pairs less than         10,000: 205
Twin prime pairs less than        100,000: 1,224
Twin prime pairs less than      1,000,000: 8,169
Twin prime pairs less than     10,000,000: 58,980
Twin prime pairs less than    100,000,000: 440,312
Twin prime pairs less than  1,000,000,000: 3,424,506
Twin prime pairs less than 10,000,000,000: 27,412,679

Phix

Added both parameter to reflect the recent task specification changes, as shown for a limit of 6 you can count {3,5} and {5,7} as one pair (the default, matching task description) or two. Obviously delete the "6 --" if you actually want a prompt.

The time complexity here is all about building a table of primes. It turns out that using the builtin get_prime() is actually faster than using an explicit sieve (as per Delphi/Go/Wren) due to retaining all the intermediate 0s, not that I particularly expect this to win any performance trophies. <lang Phix>function twin_primes(integer maxp, bool both=true) 

   integer n = 0,  -- result
           pn = 2, -- next prime index
           p,      -- a prime, <= maxp
           prev_p = 2
   while true do
       p = get_prime(pn)
       if both and p>=maxp then exit end if
       n += (prev_p = p-2)
       if (not both) and p>=maxp then exit end if
       prev_p = p
       pn += 1
   end while
   return n

end function integer mp = 6 -- prompt_number("Enter limit:") printf(1,"Twin prime pairs less than %,d: %,d\n",{mp,twin_primes(mp)}) printf(1,"Twin prime pairs less than %,d: %,d\n",{mp,twin_primes(mp,false)}) for p=1 to 9 do 

   integer p10 = power(10,p)
   printf(1,"Twin prime pairs less than %,d: %,d\n",{p10,twin_primes(p10)})

end for</lang>

Output:
Twin prime pairs less than 6: 1
Twin prime pairs less than 6: 2
Twin prime pairs less than 10: 2
Twin prime pairs less than 100: 8
Twin prime pairs less than 1,000: 35
Twin prime pairs less than 10,000: 205
Twin prime pairs less than 100,000: 1,224
Twin prime pairs less than 1,000,000: 8,169
Twin prime pairs less than 10,000,000: 58,980
Twin prime pairs less than 100,000,000: 440,312
Twin prime pairs less than 1,000,000,000: 3,424,506

Raku

Works with: Rakudo version 2020.07

<lang perl6>use Lingua::EN::Numbers;

use Math::Primesieve;

my $p = Math::Primesieve.new;

printf "Twin prime pairs less than %14s: %s\n", comma(10**$_), comma $p.count(10**$_, :twins) for 1 .. 10;</lang>

Output:
Twin prime pairs less than             10: 2
Twin prime pairs less than            100: 8
Twin prime pairs less than          1,000: 35
Twin prime pairs less than         10,000: 205
Twin prime pairs less than        100,000: 1,224
Twin prime pairs less than      1,000,000: 8,169
Twin prime pairs less than     10,000,000: 58,980
Twin prime pairs less than    100,000,000: 440,312
Twin prime pairs less than  1,000,000,000: 3,424,506
Twin prime pairs less than 10,000,000,000: 27,412,679

REXX

straight-forward prime generator

The   genP   function could be optimized for higher specifications of the limit(s). <lang rexx>/*REXX pgm counts the number of twin prime pairs under a specified number N (or a list).*/ parse arg $ . /*get optional number of primes to find*/ if $= | $="," then $= 10 100 1000 10000 100000 1000000 10000000 /*No $? Use default.*/ w= length( commas( word($, words($) ) ) ) /*get length of the last number in list*/ @found= ' twin prime pairs found under ' /*literal used in the showing of output*/ 

      do i=1  for words($);       x= word($, i) /*process each N─limit in the  $  list.*/
      say right( commas(genP(x)), 20)     @found     right(commas(x), max(length(x), w) )
      end   /*i*/

exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do ?=length(_)-3 to 1 by -3; _=insert(',', _, ?); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: parse arg y; @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13; #= 5; tp= 2 

           do j=13  by 2  while  j<y            /*continue on with the next odd prime. */
           parse var  j    -1  _              /*obtain the last digit of the  J  var.*/
           if _      ==5  then iterate          /*is this integer a multiple of five?  */
           if j // 3 ==0  then iterate          /* "   "     "    "     "     " three? */
           if j // 7 ==0  then iterate          /* "   "     "    "     "     " seven? */
           if j //11 ==0  then iterate          /* "   "     "    "     "     " eleven?*/
                                                /* [↓]  divide by the primes.   ___    */
                 do k=6  to #  while  k*k<=j    /*divide  J  by other primes ≤ √ J     */
                 if j//@.k == 0  then iterate j /*÷ by prev. prime?  ¬prime     ___    */
                 end   /*k*/                    /* [↑]   only divide up to     √ J     */
           prev= @.#;      #= # + 1;    @.#= j  /*save prev. P; bump # primes; assign P*/
           if j-2==prev   then tp= tp + 1       /*This & previous prime twins? Bump TP.*/
           end         /*j*/
    return tp</lang>
output   when using the default inputs:
                   2  twin prime pairs found under          10
                   8  twin prime pairs found under         100
                  35  twin prime pairs found under       1,000
                 205  twin prime pairs found under      10,000
               1,224  twin prime pairs found under     100,000
               8,169  twin prime pairs found under   1,000,000
              58,980  twin prime pairs found under  10,000,000

optimized prime number generator

This REXX version has some optimization for prime generation.

This version won't return a correct value (for the number of twin pairs) for a limit < 73   (because of the manner in which low primes are generated). <lang rexx>/*REXX pgm counts the number of twin prime pairs under a specified number N (or a list).*/ parse arg $ . /*get optional number of primes to find*/ if $= | $="," then $= 100 1000 10000 100000 1000000 10000000 /*No $? Use default.*/ w= length( commas( word($, words($) ) ) ) /*get length of the last number in list*/ @found= ' twin prime pairs found under ' /*literal used in the showing of output*/ 

      do i=1  for words($);       x= word($, i) /*process each N─limit in the  $  list.*/
      say right( commas(genP(x)), 20)     @found     right(commas(x), max(length(x), w) )
      end   /*i*/

exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg _; do ?=length(_)-3 to 1 by -3; _=insert(',', _, ?); end; return _ /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: arg y; _= 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 

                    #= words(_);      tp= 8     /*#: number of prims; TP: # twin pairs.*/
       do aa=1  for #;  @.aa= word(_, aa)       /*assign some low primes for quick ÷'s.*/
       end   /*aa*/

       do j=@.#+2  by 2  while j<y              /*continue with the next prime past 101*/
       parse var  j    -1  _                  /*obtain the last digit of the  J  var.*/
       if _    ==5       then iterate           /*is this integer a multiple of five?  */
       if j//3 ==0       then iterate           /* "   "     "    "     "     " three? */

          do a=4  for 23                        /*divide low primes starting with seven*/
          if j//@.a ==0  then iterate j         /*is integer a multile of a low prime? */
          end           /*a*/
                                                /* [↓]  divide by the primes.   ___    */
                  do k=27  to #  while k*k <= j /*divide  J  by other primes ≤ √ J     */
                  if j//@.k ==0  then iterate j /*÷ by prev. prime?  ¬prime     ___    */
                  end   /*k*/                   /* [↑]   only divide up to     √ J     */
       prev= @.#;          #= # + 1;     @.#= j /*save prev. P; bump # primes; assign P*/
       if j-2==prev   then tp= tp + 1           /*This & previous prime twins? Bump TP.*/
       end              /*j*/
    return tp</lang>

Rust

Limits can be specified on the command line, otherwise the twin prime counts for powers of ten from 1 to 10 are shown. <lang rust>// [dependencies] // primal = "0.3" // num-format = "0.4"

use num_format::{Locale, ToFormattedString};

fn twin_prime_count_for_powers_of_ten(max_power: u32) { 

   let mut count = 0;
   let mut previous = 0;
   let mut power = 1;
   let mut limit = 10;
   for prime in primal::Primes::all() {
       if prime > limit {
           println!(
               "Number of twin prime pairs less than {} is {}",
               limit.to_formatted_string(&Locale::en),
               count.to_formatted_string(&Locale::en)
           );
           limit *= 10;
           power += 1;
           if power > max_power {
               break;
           }
       }
       if previous > 0 && prime == previous + 2 {
           count += 1;
       }
       previous = prime;
   }

}

fn twin_prime_count(limit: usize) { 

   let mut count = 0;
   let mut previous = 0;
   for prime in primal::Primes::all().take_while(|x| *x < limit) {
       if previous > 0 && prime == previous + 2 {
           count += 1;
       }
       previous = prime;
   }
   println!(
       "Number of twin prime pairs less than {} is {}",
       limit.to_formatted_string(&Locale::en),
       count.to_formatted_string(&Locale::en)
   );

}

fn main() { 

   let args: Vec<String> = std::env::args().collect();
   if args.len() > 1 {
       for i in 1..args.len() {
           if let Ok(limit) = args[i].parse::<usize>() {
               twin_prime_count(limit);
           } else {
               eprintln!("Cannot parse limit from string {}", args[i]);
           }
       }
   } else {
       twin_prime_count_for_powers_of_ten(10);
   }

}</lang>

Output:
Number of twin prime pairs less than 10 is 2
Number of twin prime pairs less than 100 is 8
Number of twin prime pairs less than 1,000 is 35
Number of twin prime pairs less than 10,000 is 205
Number of twin prime pairs less than 100,000 is 1,224
Number of twin prime pairs less than 1,000,000 is 8,169
Number of twin prime pairs less than 10,000,000 is 58,980
Number of twin prime pairs less than 100,000,000 is 440,312
Number of twin prime pairs less than 1,000,000,000 is 3,424,506
Number of twin prime pairs less than 10,000,000,000 is 27,412,679

Wren

Library: Wren-math
Library: Wren-fmt

<lang ecmascript>import "/math" for Int import "/fmt" for Fmt

var c = Int.primeSieve(1e8-1, false) var limit = 10 var start = 3 var twins = 0 for (i in 1..8) { 

   var j = start
   while (j < limit) {
       if (!c[j] && !c[j-2]) twins = twins + 1
       j = j + 2
   }
   Fmt.print("Under $,11d there are $,7d pairs of twin primes.", limit, twins)
   start = limit + 1
   limit = limit * 10

}</lang>

Output:
Under         100 there are       8 pairs of twin primes.
Under       1,000 there are      35 pairs of twin primes.
Under      10,000 there are     205 pairs of twin primes.
Under     100,000 there are   1,224 pairs of twin primes.
Under   1,000,000 there are   8,169 pairs of twin primes.
Under  10,000,000 there are  58,980 pairs of twin primes.
Under 100,000,000 there are 440,312 pairs of twin primes.
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