Any mathematicians out there? I think I can prove the following and am wondering if it might be in any way interesting.

Proposition: Relative to the set of prime numbers below any P there are infinitely many pairs of consecutive twin primes.

It would not follow that there are infinitely many twin primes, worse luck, but still, maybe it’s of interest. Any thoughts?

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I’m not a mathematician, but I’ve toyed around with prime numbers, including failed attempts at proving Goldbach’s conjecture.

My first question is, what do you mean by “relative to any finite set of prime numbers”?

I mean that these consecutive pairs of numbers will have no factors in the finite set. I’m proposing that no finite set of prime factors could be large enough to prevent the occurrence of consecutive twin primes.

49 would be prime ‘relative’ to the primes < 7. There may be a better word than 'relative' but if so I don't know it.