Date: Jan 21, 2013 8:42 PM
Author: manontheclaphamomnibus
Subject: question about congruence classes

This following must obviously be too simple, but if someone can explain what it misses, I'd be very grateful.

If there are only finitely many pairs (n, n+2), both prime, then from the apparently infinitely many numbers x such that x is

neither [0]mod2 nor [0]mod2,
nor [0]mod3 nor [1]mod3,
nor [0]mod5 nor [3]mod5....
nor [0]modp nor [p-2](modp for all p < sqrt(x + 1)),

then above a certain size, either the seeming inductive step for filtering all candidate numbers x by the two requirements modulo the next additional prime breaks down, or else some extra condition begins to hold.

Either way, somehow even if infinitely many x remain satisfying the conditions for each specific p_i from then on, none of them after this point need be small enough to ensure any longer that infinitely many x meet these conditions for all values of p < sqrt(x + 1) in general? Is this the main problem with trying to show this?