Maybe the reason the twin prime conjecture has eluded proof is because it is false. It does not seem unreasonable that the primes eventually thin out so much that locally high density concentrations of primes stop occurring. (This would contradict the Hardy-Littlewood conjecture in "Partitio Numerorum III".)
If it is false, how would we prove it to be false? Could we find the last twin prime?
Some wider prime constellations are conjectured like twin primes to have infinite instances, but not a single example is known.
http://anthony.d.forbes.googlepages.com/ktmin.txt
Some relevant OEIS entries: A083409, A008407, A020497.
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