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

@misc{shi2026matrixtheoretic,
  title = {{A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares}},
  author = {Wujie Shi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.21529},
  eprint = {2605.21529},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.21529},
}

TY  - UNPB
TI  - A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares
AU  - Wujie Shi
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2605.21529
UR  - https://arxiv.org/abs/2605.21529
AB  - 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 even:pmin(d)≤ z} has density zero, with the asymptotic (π(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 ψ→0, pmin(d)>ψ(d)log dloglog d for almost all even d, which is the conjectured typical order; and max_d≤ Xpmin(d)≥(1/2+o(1))log Xloglog X. The same counting gives the corresponding lower bounds for the least prime in a Goldbach partition.
ER  -