For many values of n, it is faster to generate all the primes less than n than it is to generate each prime (p) up to n/2 and test whether n-p is prime. So the second routine runs faster than the first.

Reuses is_prime and primes_to from earlier problems.

from primes import primes_to, is_prime

def prime_pairs1( x ):
    return ((p,x-p) for p in primes_to(x/2) if is_prime( x-p ))

def prime_pairs2( x ):
    prime = list( primes_to( x ) )

    lo, hi = 0, len(prime)-1
    while lo <= hi:
	n = prime[hi] + prime[lo]
	if n < x:
		lo += 1
	elif n == x:
		yield prime[lo], prime[hi]
		lo += 1
	else:
		hi -= 1

mirrors(N)=
{
\\ N is the number of interest to find symmetric prime pairs
\\ written as pari/GP script
i=1;cnt=0;Q1=999;

if(N%2!=0|N5! “);return(2) );

print(“selected N = “,N);

while(Q1>5,

Q1=N-i;Q2=N+i;

if(ispseudoprime(Q1)&&ispseudoprime(Q2),
\\## print(” mirror pair at “,Q1,” / “,Q2); \\disable for speed or if N> BB1
cnt++ ); \\ end IF

i=i+2; \\ skip multiples if 5! for speed
if(i%5==0,i=i+2) \\ BB1

); \\ end WHILE

print(“# pairs found = “,cnt);
print(“other candidate N: 6,12,30,60,180,210,360,420,1260,2310,2520,4620,”);

return(0);
}

public static class Goldbach
{
private static List primes = new List() { 2, 3 };

public static void GetGoldbachPrimes(int value, out int prime1, out int prime2)
{
Debug.Assert(value > 2 && value % 2 == 0, “value > 2 && value % 2 == 0″);
if (value <= 2 || value % 2 != 0)
{
throw new ArgumentException("value must be even number greater than 2", "value");
}

foreach (var prime in GetPrimes())
{
int difference = value – prime;
if (IsPrime(difference))
{
prime1 = prime;
prime2 = difference;
return;
}
}

throw new InvalidOperationException("Primes could not be found");
}

private static IEnumerable GetPrimes()
{
for (int i = 0; i < primes.Count; i++)
{
yield return primes[i];
}

for (int i = primes[primes.Count - 1] + 2; i < int.MaxValue; i += 2)
{
if (IsPrime(i))
{
primes.Add(i);
yield return i;
}
}
}

private static bool IsPrime(int value)
{
if (value = 0)
{
return true;
}

int sqrt = Convert.ToInt32(Math.Ceiling(Math.Sqrt(value)));

for (int i = 3; i <= sqrt; i += 2)
{
if (value % i == 0)
{
return false;
}
}

return true;
}
}

#include <stdio.h>
#include <stdlib.h>

void show_usage();
unsigned long *create_sieve_to_number(unsigned long number);

int main (int argc, const char * argv[]) {

    if (argc != 2) {
		show_usage();
		return -1;
	}

	unsigned long number = atol(argv[1]);

	if ((number % 2) != 0) {
		show_usage();
		return -1;
	}

	unsigned long *sieve = create_sieve_to_number(number);

	for (unsigned long i = 2; i < number; i++) {
		if (sieve[i] == 1) {
			for (unsigned long j = i; j < number; j++) {
				if (sieve[j] == 1) {
					if (i + j == number) {
						printf("Solution found: %d + %d\n", i, j);
						return 0;
					}
				}
			}
		}
	}

	printf("no solution found!  pick up your Fields Medal!\n");

	return 0;

}

void show_usage(void) {
	printf("usage: goldbach [even number]\n");
}

unsigned long *create_sieve_to_number(unsigned long number) {
	unsigned long *sieve;

	sieve = (unsigned long *)malloc(sizeof(unsigned long) * (number + 1));

	for (int i = 0; i < number; i++) {
		sieve[i] = 1;
	}

	for (unsigned long i = 2; i < number; i++) {
		if (sieve[i] == 1) {
			for (unsigned long j = i * i; j < number; j = j + i) {
				sieve[j] = 0;
			}
		}
	}

	return sieve;
}

import Data.Numbers.Primes

goldbach :: Integer -> (Integer, Integer)
goldbach n = head [(p, n - p) | p <- takeWhile (< n) primes, isPrime (n - p)]