# Polynomials of minimum degree that interpolate primes in intervals

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).

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

• What do you mean by saying a polynomial "passes through" a prime? What points are you interpolating? – Robert Israel yesterday

Equivalently, you want to interpolate the points $$(i, p_i)$$, $$i = m \ldots n$$ where $$p_i$$ is the $$i$$'th prime.
The prime $$k$$-tuples conjecture implies that for each integer $$k > 2$$ and each $$d$$ from $$1$$ to $$k-1$$, there are infinitely many $$m$$ such that with $$n=m+k$$ the minimum degree of the interpolating polynomial is $$d$$.
The conjecture says that $$(p_{m+1}-p_m, p_{m+2}-p_m, \ldots, p_{m+k} - p_m)$$ is, for infinitely many $$m$$, any increasing $$k$$-tuple of even integers such that there is no prime $$q$$ for which this covers all residues mod $$q$$. In particular this implies that for a given $$k>2$$, if $$A$$ is the product of all primes $$\le k$$ and $$d \ge 1$$ there are infinitely many $$m$$ such that $$p_j = p_m + A (j-m)^d$$ for $$m \le j \le m+k$$. Thus in such a case, if $$d \le k-1$$ the minimum degree is $$d$$.