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

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!

• Do you mean that $n+k$ AND $n-k$ are prime or just $n+k$ OR $n-k$ is prime? – stressed out Jan 2 at 0:28
• both are primes – François Huppé Jan 2 at 0:32
• But then it's false for $n<3$. Isn't it? – stressed out Jan 2 at 0:32
• @stressed you are right – François Huppé Jan 2 at 0:35

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.
• @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. – John Omielan Jan 2 at 7:58
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."