Problems & Puzzles: Conjectures

Conjecture 62. Generalization of Bertrand's Postulate

Werner D. Sand sent the following conjecture:

For any k>0 there is an N(k) such that for all n>N applies: In the interval [n,2n] there exists (at least) one k-tuple of prime numbers (prime constellations).

I checked it for:

k=2 (twins), N=6,
k=3 (triples), N=21,
k=4 (quadruples), N=414,
each with n up to 20000

Q1. Can you prove or disapprove this Conjecture by Sand?

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