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I have read quite a bit about prime numbers recently (having just started a module on elementary number theory, groups, primes, etc.), and something that always seems to be popping up is powers of 2. I am reasonably familiar with Mersenne primes, and have seen a few other reasons why $2^n\pm1$ keeps on appearing (especially to do with some computing things as well, it seems).

Like pretty much every other maths lover who has come in to contact with primes and their mysteries, I've played around a bit and tried writing programs to produce primes in different ways, or just pondered their (potential?) pattern.

Anyway, to the point:

Do we know anything (well, maybe not anything before anybody points that out, but anything 'sufficiently' interesting I guess) about what conditions $x$ must satisfy in order for $2x+1$ to be prime?

(Potentially) further:

Analogously, for $3x\pm1$, or $5x\pm\{1,2\}$ (not too sure if that notation is acceptable?), $7x\pm\{1,2,3\}$, etc.

I guess you could also talk about $4x+\{1,2,3\}$, but that seems pretty similar to discussing the $2x+1$ case. Really I see how this whole thing might be a bit pointless to discuss, as the cases you consider really depend on primes themselves, so the recursive behaviour of these conditions make them no more useful than knowing that for a number to be prime it can't divide by any of the primes below it: i.e. you still need to know the primes below it!

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There are also factorial primes of the form $n! \pm 1$. – amWhy May 6 at 0:44
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If $a$ and $b$ are relatively prime, with $a\ne 0$, there are, by a theorem of Dirichlet, infinitely many primes of the form $ak+b$. So one can say very little about primes of the form $ak+b$, except that everything which is not obviously impossible is in fact possible. – André Nicolas May 6 at 1:36

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Better grab some text on number theory (like Moser's "An introduction to the theory of numbers"). One infuriating particularity of number theory is that any Goldbach can come along and state that each even number larger than 4 is the sum of two prime numbers, any bright fifth grader can understand that and check the first few examples, while stumping centuries worth of first-rate mathematicians.

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