Why are conjectures about the primes so hard to prove?

Solution 1:

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.

Solution 2:

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.

• Link $1$
• Link $2$
• Link $3$

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