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How to prove the following conjecture:

For any positive integer $n>3$, there exists at least $1$ integer $k$ such that $n+k$ and $n-k$ are both primes.

Any hint, idea or reference would be greatly appreciated!

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  • $\begingroup$ Do you mean that $n+k$ AND $n-k$ are prime or just $n+k$ OR $n-k$ is prime? $\endgroup$ – stressed out Jan 2 at 0:28
  • $\begingroup$ both are primes $\endgroup$ – François Huppé Jan 2 at 0:32
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    $\begingroup$ But then it's false for $n<3$. Isn't it? $\endgroup$ – stressed out Jan 2 at 0:32
  • $\begingroup$ @stressed you are right $\endgroup$ – François Huppé Jan 2 at 0:35
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I suspect the OP means $n + k$ AND $n - k$. If so, this is the same as the strong Goldbach conjecture, which is a very well-known unsolved problem. In particular, if $n + k$ and $n - k$ are both prime, their sum is $2n$. Also, every positive even integer $\gt 2$ is of the form $2n$, for some positive integer $n$. Thus, if the OP's conjecture holds, so does the Goldbach conjecture.

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  • $\begingroup$ @FrançoisHuppé I am not quite sure what you are stating. I never wrote that, or meant to imply, it was "trivial". I just explained that it is a slightly stronger conjecture than the strong Goldbach conjecture. I was not aware from your question that you already knew this, and I didn't mean to imply you did not because there is no way for me to know what you have already figured out apart from what you wrote. $\endgroup$ – John Omielan Jan 2 at 0:49
  • $\begingroup$ sorry it came out wrong, i should add this in my question instead :) in fact your answer is perfect $\endgroup$ – François Huppé Jan 2 at 0:58
  • $\begingroup$ @FrançoisHuppé No worries about your earlier comment. Also, I originally thought your question said $k$ is a positive integer. Perhaps it did then but was changed since as I see the question has an edit time after my answer. Regardless, as the question text now does not require this, it means your conjecture is the same as the Goldbach one, so I have changed my answer accordingly to no longer say yours is a slightly stronger conjecture. $\endgroup$ – John Omielan Jan 2 at 7:58
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I would do some searching in the On-Line Encyclopedia of Integer Sequences (OEIS). Give me a couple of minutes...

Okay, here's one pertinent result: http://oeis.org/A020483 Least prime $p$ such that $p + 2n$ is also prime. According to Jens Kruse Andersen, "It is conjectured that $a(n)$ always exists."

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