@Vassilis, I think we have established that your question is not suitable to MathOverflow.
]]>You have the right to believe the cause is complexity of the question. I have the right to believe that it is matters of exposition and lack of clarity that are responsible for my and for others not understanding your question. I will bet a small amount of money that the people you are interested in having understand the question will face difficulties for the reasons I believe more than for the reasons you believe.
I stand by my earlier points regarding communication and question asking. Likewise for the last two paragraphs of my earlier post.
Gerhard Paseman, 2011.11.18
]]>" I have come across the following pattern. For small n, I am finding (n primes or n consecutive primes, you make clear which) q_i all of the form 4k+1, such that Lower(n) < q_i < Upper(n) . (I'll let you define Lower and Upper in terms of 5^(2^(n-1)) .) Does this hold for all sufficiently large positive integers n? "
As written, this is a question which is barely suitable for MathOverflow, primarily because the motivation for finding such a constellation is unclear. However, it has the benefit of (after adding the defining details) being clear and precise and understandable by others. Your previous posts did not meet the standards of clarity I would insist upon for a MathOverflow question.
Now that I understand the challenge, I will repeat the advice others have given. This question is more appropriate for math.stackexchange. The process of asking for and getting help on forming the question is better suited for meta.math.stackexchange. I have not done the calculations, but if you do them, I think you will find that the result is an easy consequence (even for consecutive primes) of the prime k-tuples conjecture, and that you will have more like O(2^(polylog(n)) sequence of primes in the desired interval.
I will say this one last time: Ask for help (in writing the question down) on meta.math.stackexchange or in some other forum; Ask the question itself on math.stackexchange. If you get no response, ask (on meta.math.stackexchange) for help posting to MathOverflow. Do not post such questions on MathOverflow yourself. Do not ask for help on meta.matheoverflow yourself. ONLY IF YOU GET HELP FROM OUTSIDE MATHOVERFLOW by someone who can help you form a good question, only then should you post on MathOverflow, and say who helped you.
I don't want to stop your research or your pursuit of mathematics. I encourage you to ask good questions on the right forums. The past few attempts have convinced me that you need more practice on places outside of MathOverflow: you are not ready to post on MathOverflow.
I wish you well.
Gerhard "Ask Me About System Design" Paseman, 2011.11.18
]]>I have left some comments on your question. But the short answer is: I don't understand at all what you're asking, and in order for the question to be reopened it's vitally important that at least the question be clearly stated in standard mathematical language.
]]>the fact that this question is of fundamental importance concerning the distribution of primes
Is this in the same sense that it's a fact that there are nine million bicycles in Beijing?
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