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

Abstract

Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, the number of times $n$ appears in $B$. We prove the exact identity \[ P_k = N_k - S_k + E_k \] where $P_k = \#\{\text{primes in } I_k\}$, $N_k = 4k$ counts the odd integers in $I_k$, $S_k = \sum_{n \in I_k \text{ odd}} r(n)$ is the total matrix multiplicity, and $E_k = \sum_{n \in I_k \text{ odd}} (r(n)-1)$ measures the excess multiplicity of non-semiprime odd composites. All three quantities $N_k$, $S_k$, $E_k$ are computable from the divisor structure of odd integers in $I_k$ without primality testing. The formula yields the equivalent combinatorial condition: \[ P_k \geq 1 \iff E_k \leq S_k - N_k. \] We verify $P_k \geq 1$ for all $k \leq 10^8$ by direct computation and establish $P_k \geq 1$ for all $k \leq 1.37 \times 10^{17}$ using the Baker-Harman-Pintz theorem [2]. Whether $P_k \geq 1$ for all $k$ (a weaker statement than Legendre's conjecture) remains an open problem, now equivalent to the purely combinatorial inequality $E_k \leq S_k - N_k$ for all $k$.