4 problems

Conjecture. There are infinitely many primes $p$ such that $p+2$ is also prime: …

Conjecture. For every admissible family, there are infinitely many positive integers $n$ such that $f_1(n),\dots,f_k(n)$ are all prime.

Conjecture. For every integer $n\ge1$ there is a prime strictly between $n^2$ and $(n+1)^2$: …

Conjecture (Elliott–Halberstam). The primes have level of distribution $\theta$ for every $\theta<1$. That is, for every fixed $\theta<1$ and every $A>0$ there is a constant…