Why are conjectures about the primes so hard to prove? I recently started learning number theory, and I've noticed there are many conjectures about the prime numbers that are unproven. Some examples would be whether there are infinite Mersenne, Sophie-Germain, or Fermat primes. There are also problems like the Goldbach conjecture. Yet, primes are defined so simply. Why are they so hard to work with? I am very new to number theory, so I am looking for a simple, intuitive answer.
 A: I'm surprised that the other answers have not touched on the distribution of prime numbers.  It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation.  This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers.  This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$.  Terence Tao has a lot of literature on this directed toward the layperson.

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*Link $1$

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*Link $3$
In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.
A: The prime numbers are an odd set of numbers.
The crux of the problem lies in the following fact:
the function $P(n)$ that outputs the n'th prime is still only known to be defined using the idea "the nth prime is not divisible by any primes below it" and "the first prime is 2". This sort of definition means any question involving prime numbers cannot be solved easily. An example is the twin prime conjecture. The question quite simply asks:
are there infinitely many n such that
$$P(n) - P(n-1) = 2?$$
If we had a non-recursive (ie an explicit formula no matter how complicated) for describing the function then suddenly many many tools like calculus (think Newton's Method), functional analysis, theory of solvability of equations etc... can be used to tackle this problem.
But WE DONT HAVE AN EXPLICIT DEFINITION. 
So all of that goes out the door and is essentially what makes Number Theory so brutally challenging yet so wonderfully rewarding.
A: Here's a brief, vague, and incomplete answer but one that relates to many open problems (and both the Goldbach and Twin Prime conjectures): primality is a multiplicative condition, not an additive one. Understanding how they are distributed in an arithmetic sequence (e.g. in the natural numbers or for use in problems involving prime gaps (roughly what's needed in the above two conjectures)) is attempting to understand how the notion of primality relates to something defined in terms of addition, and there's no obvious way to understand the definition of a prime number in terms of addition.
Edit: I just noticed that this was already mentioned in the comments but may as well leave it here. frogeyedpeas's answer brings up another very good point.
