Define a chain of primes of length n: {p_1,p_2...,p_n} such that

C(p_(k-1))=p_(k)+1 for all 2<=k<=n where C(p_(k-1)) is the p_(k-1)-th composite.

Example: {197,251,317} is a chain of length 3 because C(197)=252=251+1

and C(251)=318

Another property is: p_(k)=P(s)=s-th prime--->p_(k-1)=P(s)-s

There are 120 chains of length 4 for p_4<5*10^6

4 chains of length 5 for p_5<5*10^6

1 chain of length 6 for p_6<5*10^6

The first two n=5 chains are: {11,19,29,43,61} and {142573,157007,172741,189901,208589}

The first two n=6 chains are: {7829,8941,10193,11587,13151,14897} and {4523081,4862593,5225683,5613763,6028483,6471431}

A very rough heuristic seems to indicate that the first p-10 would be a 15 digit prime.

Computation is here very intensive.

Would anybody try to find a fast (say PARI script)?

And a good heuristic?

Find a chain for n=7 and 8

Yet another sets of simoultaneous primes,Jens.

Where (number of digits) do you think we would find a 19-chain and WHEN?

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