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

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The article presents a matrix-theoretic exact formula to count primes in intervals between consecutive odd squares, defining matrix multiplicity and proving an identity that calculates the number of primes in each interval without primality testing. The formula confirms at least one prime exists in such intervals up to $1.37 \times 10^{17}$, establishing a combinatorial condition equivalent to this property.

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A new formula for counting primes in intervals between consecutive odd squares has been derived using matrix theory.

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arXiv:2605.21529v2 Announce Type: replace 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 $k$.