TITLE: Why are conjectures about the primes so hard to prove?
QUESTION [18 upvotes]: 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.

REPLY [8 votes]: 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 layman.

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In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.