<PRE><B>Question 169428</B>
First of all, you have misstated Goldbach&#39;s conjecture.  The conjecture, properly stated, is:


&lt;i&gt;Every&lt;/i&gt; &lt;b&gt;&lt;u&gt;even&lt;/u&gt;&lt;/b&gt; &lt;i&gt;integer greater than 2 can be expressed as the sum of two primes.&lt;/i&gt;


First thing is to define &#39;counterexample&#39;  A counterexample is an example that disproves the conjecture.  A simple illustration would be thus:  Let&#39;s say that you came to me with the conjecture that &quot;All dogs are black.&quot;  I could then disprove your conjecture by finding a white dog (or a brown one, for that matter).


A counterexample to the Goldbach conjecture would be to find an even integer (the counterexample number) where &lt;i&gt;&lt;b&gt;every&lt;/i&gt;&lt;/b&gt; prime number less than the counterexample number when subtracted from the counterexample number results in a non-prime difference.


So, if {{{N}}} (even and greater than 2) is the counterexample number, and { {{{x[1]}}},{{{x[2]}}},{{{x[3]}}},...,{{{x[p]}}} } is the set of &lt;i&gt;&lt;b&gt;ALL&lt;/i&gt;&lt;/b&gt; prime numbers less than {{{N}}}, then &lt;i&gt;&lt;b&gt;NONE&lt;/i&gt;&lt;/b&gt; of the subtractions {{{N - x[i]}}} (where i =  1 through p) results in a prime number.


The strong Goldbach conjecture has never been proven in general, but neither has a counterexample ever been found.  If you want to go off on your own search for an actual counterexample to Goldbach, start with numbers greater than {{{10^18}}} because every possibility less than that has already been shown to fit the conjecture.</PRE>