A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares

Abstract

Matrix $B=(b_{ij})$ with $b_{ij}=(2j+1)(2j+2i-1)$ was introduced in \cite{Shi2024} as an additive sieve for odd primes. In this paper we introduce the minimal-anchor function $\pmin(d)$, the least odd prime $p$ such that $p+d$ is prime (sequence A020483 of the OEIS at index $d/2$), whose finiteness for all even $d$ is exactly the weak Polignac (Maillet) conjecture, i.e.\ the statement that every row of $B$ contains a semiprime. We prove that for each fixed $z$ the set $\{d\ \text{even}:\pmin(d)\le z\}$ has density zero, with the asymptotic $(\pi (z)-1)X/\log X$; consequently no fixed finite set of anchor primes can cover a positive proportion of the rows. Density-one coverage of the rows nevertheless holds, by a classical theorem of Lavrik which we restate in the matrix-$B$ framework: almost every row contains the number of semiprimes predicted by the Hardy--Littlewood conjecture. We formulate quantitative conjectures on $\pmin$ and support them with numerical data. Every-row coverage (= weak Polignac) is explicitly left open. We prove unconditional lower bounds for $\pmin$: for every $\psi \to 0$, $\pmin(d)>ψ(d)\log d\log\log d$ for almost all even $d$, which is the conjectured typical order; and $\max_{d\le X}\pmin(d)\ge(\frac12+o(1))\log X\log\log X$. The same counting gives the corresponding lower bounds for the least prime in a Goldbach partition.

13 pages