Twin primes
Twin primes are pairs of natural numbers (P1 and P2) that satisfy the following:
- P1 and P2 are primes
- P1 + 2 = P2
- Task
Write a program that displays the number of twin primes that can be found under a user-specified number.
- Examples
> Search Size: 100 > 8 twin prime pairs.
> Search Size: 1000 > 35 twin prime pairs.
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.
REXX
The genP function could be optimized for higher specifications of the limit(s).
Note that this REXX program counts the number of twin primes, not the number of twin pairs. <lang rexx>/*REXX program counts the number of twin primes under a specified number N (or a list).*/ parse arg $ . /*get optional number of primes to find*/ if $= | $="," then $= 100 1000 10000 100000 /*Not specified? Then assume default.*/ w= length( word($, words($) ) ) /*get length of the last number in list*/
do i=1 for words($); x= word($, i) /*process each N─limit in the $ list.*/ say right( genP(x), 20) ' twin primes found under ' right(x, max(length(x), w)) end /*i*/
exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: arg y; @.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13; #= 5; tp= 3; s= @.# + 2
do j=s by 2 while j<y+2 /*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 */ #= #+1 /*bump the count of number of primes. */ @.#= j; _= # - 1 /*define J prime; point to prev. prime.*/ if j-2\==@._ then iterate /*This & previous prime not twins? Skip*/ if j-2<y then tp= tp + 1 /*Is this part of the twin pair? Bump.*/ if j <y then tp= tp + 1 /* " " " " " " " " */ end /*j*/ return tp</lang>
- output when using the default inputs:
15 twin primes found under 100 69 twin primes found under 1000 409 twin primes found under 10000 2447 twin primes found under 100000