# Upper bound for the number of $k$-central numbers in a prime gap

Let $$I_{n}:=]p_{n},p_{n+1}[$$ be the open interval between the $$n$$-th and $$(n+1)$$-th prime. Under Goldbach's conjecture, denote by $$r_{0}(m)$$ the smallest positive integer $$r$$ such that both $$m-r$$ and $$m+r$$ are prime for any large enough composite integer $$m$$ and by $$k_{0}(m)$$ the quantity $$\pi(m+r_{0}(m))-\pi(m-r_{0}(m))$$. Any integer $$m$$ such that $$k_{0}(m)=k$$ will be called a $$k$$-central integer and $$k$$ its order of centrality.

There is exactly one $$1$$-central integer $$m$$ in $$I_{n}$$, namely $$\frac{p_{n}+p_{n+1}}{2}$$. Moreover, letting $$l(n)$$ be $$\sup_{m\in I_{n}}\{k_{0}(m)\}$$ one can expect that the number of $$k$$-central integers in $$I_{n}$$, denoted by $$N_{I_{n}}(k)$$, is upper bounded by some constant $$C_{k}$$.

Can one prove one has $$N_{I_{n}}(k)\leq k$$ for all $$1\leq k\leq l(n)$$?

In that case the prime gap $$g_{n}:=p_{n+1}-p_{n}$$ would fulfill $$g_{n}\leq 1+\frac{l(n)(l(n)+1)}{2}$$.

Edit: the following link Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$? may help upper bound $$l(n)$$ in terms of $$n$$.

Edit: say $$g_{n}$$ is an $$l$$-grade prime gap if $$l(n)=l$$. Can one find an upper bound for every $$l$$-grade prime gap depending only on $$l$$?

I thus propose the following conjecture: Grade conjecture $$\forall l>0$$, there are infinitely $$l$$-grade prime gaps.

Note that for $$l=1$$, we recover the twin prime conjecture.

$$N_{I_n}(k)\leq k$$ indeed always holds. Let $$a_1<\dots be $$m$$ elements in $$I_n$$ such that $$k_0(a_i)=k$$ for each $$k$$. Let $$b_i=a_i-r_0(a_i),c_i=a_i+r_0(a_i)$$ so that $$b_i,c_i$$ are primes and there are exactly $$k-1$$ primes strictly between $$b_i$$ and $$c_i$$ for each $$i$$.
I claim $$b_1<\dots. Indeed, if $$b_i>b_j$$ for some $$i, then from $$a_i we deduce $$c_i=2a_i-b_i<2a_j-b_j=c_j$$. But then we find there are more primes between $$b_j,c_j$$ than between $$b_i,c_i$$, hence the claim.
Finally observe $$b_m\leq p_n. Since there are $$k-1$$ primes between $$b_1$$ and $$c_1$$, and $$b_2,\dots,b_m$$ all lie in this interval, we get $$m\leq k$$.
• Assuming the prime tuple conjecture, the bound $N_{I_n}(k)\leq k$ will also be optimal for every $k$. I doubt an unconditional proof is possible here though. Mar 2, 2021 at 12:57
• I have no idea what relation whatsoever to Cramer's conjecture this bears. If you are intending to use the bound in terms of $l(n)$ you propose, then I doubt it given there is no way to derive any nontrivial bounds from Goldbach conjecture. Mar 2, 2021 at 14:55
• Goldbach conjecture doesn't guarantee anything beyond $l(n)=O(n)$. True order of growth is probably smaller but we would need stronger assumptions to prove that. Mar 2, 2021 at 15:10