2 problems

Sun's conjecture. The least such $m$ is the first prime $p\geq 2n-1$ of the specified form $x^2+x+1$ or $4x^2+1$, respectively.

Sun's conjecture. The least positive integer $m$ for which both sets have cardinality $n$ is the least prime $p\geq 2n-1$ such that $p+2$ is also prime.