Given an interval $[a,b]$ what is the minimum degree of univariate polynomials in $\mathbb Q[x]$ that passes through all primes between $a$ and $b$ (denoted by $\mathbb P[a,b]$ with total number of primes in $[a,b]$ given by $\pi(b-a)=|\mathbb P[a,b]|$)?

$\forall x,y\in\{1,\dots,\pi(b-a)\},f(x),f(y)\in\mathbb P[a,b]$ and $f(x)<f(y)\iff x<y$.

It is at most $O(\pi(b-a))$ and is there a reason to believe it cannot be $o(\pi(b-a))$?