Euclid's proof of the infinity of primes basically works because the symmetrical pattern built up by the primes must leave a gap for any finite set. This gap must be filled by a higher prime than the set we were considering (or a multiple of such a prime).

This works because it means that we need a higher prime to fill the gap.

In the case of twin primes, the existence of the gap in the pattern simply means that we need a higher

**prime**, not a higher pair of twin primes.

So the proof would additionally need to show that you can't close all the gaps without moving up through the sieve to ever higher

**twin primes**- otherwise it doesn't work.

I was grateful to him for spotting the problem and explaining it patiently.

So a failure, but I learnt something along the way, if only how difficult this problem is.

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