In 1850 P.L. Chebyshev and in 1932 P. Erdos proved that [n,2n] interval contains a prime. Bachraoui proved the case k = 2 in 2006. k=3 case proved by Andy Loo in 2011.
From the other hand Legendre’s Conjecture states that for any n, there is a prime in [n2, (n+1)2]. For instance between 132 (=169) and 142 (=196) there are five primes (173, 179, 181, 191, and 193).
This interests me and I formulated two different questions. I hope that the problems are interesting for someone else
There is an uncertainty in the formulation of the problem : n > 1 condition should be included , i.e. interval [8,9] (and other similar intervals) can not be considered anymore.
About the rest - I have no idea
Dear Peter thanks for useful and stimulating discussion.
Bertrand’s postulate states that for every positive integer n, there is always at least one prime p such that n < p < 2n.
A generalization of the problem asked is whether for all integer n > 1 and a fixed integer k
What is the meaning of the question? It is to ask and learn, particularly it is useful when You are not expert of the field. I had a question and asked, this is first what I would like to clarify. Then I am talking about problems formulated and treated before we were born, namely I am talking about Legendre's conjecture (http://en.wikipedia.org/wiki/Legendre%27s_conjecture) and Bertrand's postulate (http://en.wikipedia.org/wiki/Bertrand%27s_postulate, http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.4451v2.pdf).
Next if You have examples being counterexamples then (as You are not expert as well) we could contact with experts to have their opinion, but we can check to see if Your examples are really counterexamples of the problems (within the links) or not? For my questions they can not be solutions. We already clarify this.
About your question which I did not answer properly- I will left it up to the moment when an expert will join to the conservation and will give a hint.
Recently Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao have produced a paper that proves a lower bound for the maximal gap
G(X) := \sup_{p_n, p_{n+1} \leq X} p_{n+1}-p_n
between primes bounded by X of the form
G(X) \geq c \log X \frac{\log \log X \log\log\log\log X}{\log\log\log X}
for some small constant c.
see: http://arxiv.org/abs/1412.5029 and the as always excellent blog post by Terrence Tao
http://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/
Yes, the conjecture is true, read my paper, still under review (communicated to nntdm) but it is available in www.vixra.org (also, see googlescholar). I hope my proof may be correct.
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