@MicrmSo you are assuming it is false now?

So the logic of your proof is:

Assume twin prime conjecture is false.

Then there is one prime doing the last elimination.

If there is such a prime, then the conjecture is false.

Therefore it's true.

Great logic!

I think that you are not quite following what his argument boils down to.

I think what he originally intended (but did not say) was this syllogism:

(1) If there exist unique integers m and u such that for any integer n $\ge$ m > 0 u either (6n - 1)/u or (6n + 1)/u is an integer, then the conjecture is false.

(2) No such m and u exist.

(3) Therefore the conjecture is true.

The first proposition is self-evidently true because it would entail that at least one member of each 6n - 1 and 6n + 1 pair was composite. I suspect that the second proposition is also true but did not attempt a proof because even if both propositions 1 and 2 are true, proposition 3 does not follow as a consequence. The syllogism is a well known fallacy.

I may have misunderstood what he was originally trying to say. In any case, he seems now to have revised his argument.

(1a) Only if there exists unique integers m and u such that, for any integer n $\ge$ m, either (6n - 1)/u or (6n + 1)/u is an integer will the conjecture be false.

(2a) No such m and u exist.

(3a) Therefore the conjecture is true.

THAT is a valid syllogism. But proposition (1a) is no longer self-evidently true, and no proof has been forthcoming except hand-waving about "last" and "final" in the context of infinite sets. In fact, I suspect proposition 1a is false.

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