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As in the title, for which positive integers $k$, $3k+1$ is prime? I think that $k$ at least cannot be $k=2m+1$, because then $3(2m+1)+1=6m+4=2(3m+2)$, which is not a prime number, so we got to look at even numbers $k$.

It also cannot be of form $k=10m+8$, because then $3(10m+8)+1=30m+25=5(6m+5)$, which is not a prime number.

Any ideas?

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    What type of answer are you looking for? A condition on $k$ such that $3k+1$ is prime? – 2017-01-05
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    Was this task assigned to you or did you think of it? Because it seems rather hard :) – 2017-01-05
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    There is no evident pattern to give a satisfactory answer. This question is as hard as "which positive numbers are prime?", which is somehow unsolved. – 2017-01-05
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    To be exact, I'm looking for such set $A$, that "$k \in A \Leftrightarrow p \text{ is prime}$". – 2017-01-05
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    You can find few elements of $A$ like $k=2,4,6,8,10,12,14,20$.But the whole set $A$ is not known. – 2017-01-05

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We do know that there are infinitely many primes of the form $3k+1$ by Dirichlet's theorem on primes in arithmetic progressions, but we do not know how this set looks like. In fact, we do not even know explicitly how the set of all primes looks like, i.e., taking the progression $a_k=k$ instead of $a_k=3k+1$ we cannot give a formula to say that $k\in A \Leftrightarrow p \text{ is prime}$. We can say, it starts with $k=2,3,5,7,11,13,\ldots,$ and there are infinitely many of them, and locally they behave very irregular, but globally they behave very regular, with the law $k/\log(k)$ etc.