Why GPY result is not a proof of the Twin Prime Conjecture?

[Guest]

The paper published by Goldston, Pintz, Yildirim [https://arxiv.org/pdf/math/0508185.pdf] proves that "that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing" (see Abstract).

This article in Quanta Magazine (https://www.quantamagazine.org/yitang-z ... -20130519/) says:

"Instead, it showed that there will always be pairs of primes much closer together than the average spacing predicts. More precisely, GPY showed that for any fraction you choose, no matter how tiny, there will always be a pair of primes closer together than that fraction of the average gap, if you go out far enough along the number line. But the researchers couldn’t prove that the gaps between these prime pairs are always less than some particular finite number."

Can someone explain why this is so? Why would an arbitrarily small fraction of the average gap not include 2?

[joe_peter]

The paper published by Goldston, Pintz, Yildirim [https://arxiv.org/pdf/math/0508185.pdf] proves that "that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing" (see Abstract).

This article in Quanta Magazine (https://educatoroid.com/is-finite-math-hard/) says:

"Instead, it showed that there will always be pairs of primes much closer together than the average spacing predicts. More precisely, GPY showed that for any fraction you choose, no matter how tiny, there will always be a pair of primes closer together than that fraction of the average gap, if you go out far enough along the number line. But the researchers couldn’t prove that the gaps between these prime pairs are always less than some particular finite number."

Can someone explain why this is so? Why would an arbitrarily small fraction of the average gap not include 2?

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