yspm's blog

A Question about Prime
By yspm, history, 2 years ago, In English

I met a problem recently which need to prove the below conclusion.

$$$\exists B,\forall n\ge B,\exists p>\sqrt n, \text{p is prime,(2p+1) is prime}$$$

There is a example that $$$n=13,p=5,2p+1=11$$$ meet this constraint

I wrote something and check it's correct when $$$40\le n\le 10^5$$$ but is there any complete proof for such B exists?

I've learnt Bertrand-Chebyshev Theorem that there exists a prime $$$p\in[n,2n]$$$ and it may help.

Thank you.

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2 years ago, # |
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There is no complete proof; it would imply that there are infinitely many Sophie-Germain primes, which is an unproved conjecture as of now.

Wikipedia: Sophie Germain primes

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2 years ago, # |
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Though I don't know how to prove or disprove this, the conclusion recalls me 102055C - GCD Land. My solution requires finding $$$p \in (\sqrt{n}, n)$$$ and $$$k \geq 2$$$ such that $$$p\ \text{is prime}, (kp+1)\ \text{is prime}$$$ and $$$(k+1)p+1 < n$$$. Such $$$p$$$ and $$$k$$$ exist for $$$35 \leq n \leq 10^5$$$, but I don't know their existence for larger $$$n$$$ either. Maybe this would be easier to prove.

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    2 years ago, # ^ |
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    I wonder whether the problem setter gives the proof? Or your solution is different from the standard one?

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