Composites and Primes
- It is not very hard to prove by induction that n_th composite number C(n) is:
C(n)=n+k where k is such that nextprime(C(n))=k-th prime.
If we choose now n=p=prime and k =C(p)-p=prime ,we obtain the following sequence of composites with a prime index and nextprime(C(p))=(C(p)-p)-th prime with C(p)-p prime.------->8,10,14,30,54,58,62,66,82,108,114,120,178,182,204,210,318,324,330,352,
So C(37)=54=37+17 and P(17)=59=nextprime(C(37))
We get here a relaton composite-prime that do not depend on a divisibility criteria.Thouh knowing n_th composite still depends of the knoledge of primes.
And we get even composites that are sum of 2 primes.(not all even composites,of course)
I cannot prove this sequence is infinite,though Mike or Jens might manage it.
If a proof of infiniteness of this sequence is considered to be too hard,then what can we say of a proof of Goldbach conjecture???
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