They just look for counterexamples. Say, for Goldbach:<p><pre><code>  is_prime(n) {
    for (i = 2; i &lt; n; i++)
      if (n % i == 0)
        return false;  &#x2F;&#x2F; sorry, this isn&#x27;t prime
    return true;  &#x2F;&#x2F; no divisors found, prime
  }

  main() {
    for (i = 4; ;i += 2) {  &#x2F;&#x2F; consider all even numbers i &gt; 2
      for (j = 2; j &lt;= i-2; j++) {  &#x2F;&#x2F; consider i as the sum j + (i-j)
        if (is_prime(j) &amp;&amp; is_prime(i-j))
          break;  &#x2F;&#x2F; both are prime, so i is a sum of two primes
      }
      if (j &gt; i-2) &#x2F;&#x2F; oops, we checked all js and nothing was found
        return i;  &#x2F;&#x2F; i is an even number but not a sum of two primes
    }
  }
</code></pre>
You can compile such program yourself and run it to see if it stops ;)<p>Just remember to use some bignum library instead of plain <i>ints</i>.