Is every even number greater than 4 the sum of a prime number with another number that belongs to the set of twin primes?
Assume the Twin Prime conjecture is True, It's equivalent to there being infinitely many numbers of form $12m$ that have a Goldbach Partition $(6m-1,6m+1)$. Now consider what happens to the values of 3 times the sum of the values of $m$, they get 2 Goldbach partitions for $6m$ equidistant for the higher value. Only when the sums of $m$ values are even, can this continue with 3 times the sum of all $m$ used (with duplication possible) over 2 when it uses 4 indices. So the conjecture, relies on the density of the indices $m$ that sum to even values, being dense enough.