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 $k$.

• The paper introduces an exact matrix-theoretic formula for counting primes in intervals defined by consecutive odd squares.
• It leverages an odd-composite matrix to encode divisor information and eliminate the need for explicit primality testing.
• The approach reframes classical prime distribution challenges, offering new insights into Legendre’s conjecture and sieve limitations.

Matrix-Theoretic Formulation of Prime Distribution in Odd Square Intervals

Introduction

This work introduces a novel, exact matrix-theoretic formula for enumerating primes in intervals between consecutive odd squares, specifically intervals of the form $I_k = ((2k - 1)^2, (2k + 1)^2)$ for integers $k \geq 1$ (2605.21529). The proposed approach leverages an 'odd-composite matrix' $B = (b_{ij})$ previously introduced by the author, wherein the occurrence multiplicity $r(n)$ of an odd integer $n$ within $B$ encodes critical divisor information sufficient to enumerate primes in $I_k$ entirely via combinatorial and arithmetic characteristics, without recourse to explicit primality testing. This framework reframes classical short-interval prime-existence problems, situating open conjectures such as Legendre's within a novel combinatorial context.

Matrix $B$ and the Multiplicity Function

The matrix $B$ is defined by $b_{ij} = (2j + 1)(2j + 2i - 1)$ for positive integers $k \geq 1$0, generating entries that exhaust all odd composites at least once. Crucially, an odd integer $k \geq 1$1 is prime if and only if $k \geq 1$2 is absent from $k \geq 1$3. For each odd $k \geq 1$4, the matrix multiplicity $k \geq 1$5 is the count of all odd divisors $k \geq 1$6 with $k \geq 1$7. The classification derived from $k \geq 1$8 is:

• $k \geq 1$9 if and only if $B = (b_{ij})$0 is an odd prime
• $B = (b_{ij})$1 if and only if $B = (b_{ij})$2 is an odd semiprime ($B = (b_{ij})$3 for odd primes $B = (b_{ij})$4)
• $B = (b_{ij})$5 for odd composites with more than two (not necessarily distinct) odd prime factors

All relevant quantities $B = (b_{ij})$6, $B = (b_{ij})$7, and $B = (b_{ij})$8 for each interval $B = (b_{ij})$9 are directly computable based solely on odd divisor structure, fully circumventing explicit primality testing.

The Main Formula for Prime Counting in $r(n)$0

The principal result is the exact formula:

$r(n)$1

where:

• $r(n)$2: number of primes in $r(n)$3
• $r(n)$4: number of odd integers in $r(n)$5
• $r(n)$6: sum of all matrix multiplicities $r(n)$7 for odd $r(n)$8 in $r(n)$9
• $n$0: aggregate excess multiplicity of non-semiprime, odd composites in $n$1

This identity redistributes the standard prime counting task onto the combinatorics of divisibility and multiplicity within the matrix $n$2. Its proof is grounded in decomposing the population of $n$3 according to the values of $n$4 and establishes that $n$5 is fully characterized by these multiplicity classes. Notably, this approach eliminates the parity problem inhibiting classical sieve methods: the combinatorics of $n$6 sidestep the indistinguishability of primes and semiprimes inherent in linear sieves.

Equivalent Combinatorial Criterion and Asymptotics

The formula immediately yields the equivalence:

$n$7

Thus, the existence of a prime in $n$8 is encoded as a condition on the excess coverage by higher-order composites relative to overcounting in $n$9 due to non-unique factorizations.

Asymptotically, the difference $B$0 grows like $B$1 and $B$2 tracks closely behind, with their ratio empirically converging to 1 from above as $B$3. Direct computation confirms $B$4 for all $B$5 and, via the Baker–Harman–Pintz theorem on prime gaps, for $B$6 up to approximately $B$7.

Implications and Connections to Existing Theory

The paper situates its results vis-à-vis notable open problems. The assertion $B$8 is weaker than Legendre's conjecture, which requires at least one prime in each of the two subintervals forming $B$9. Nevertheless, the exact formula and reformulation in terms of $I_k$0's combinatorics present a potentially tractable approach to these classical conjectures by shifting the analytic difficulty into purely arithmetic territory.

Inclusion–exclusion expansion of the main formula exposes large oscillations, emblematic of the parity problem in sieves, yet the matrix formulation remains robust and bypasses this obstacle analytically.

The matrix $I_k$1 and its multiplicity structure offer a new lens for addressing prime existence in short intervals beyond the range of classical sieve methods. The framework may have implications for further refining lower bounds on prime densities in quadratic and marginally shorter intervals, as well as for computational approaches to testing variants of Legendre-type problems.

Open Problems and Future Directions

Major unresolved questions remain, most notably whether $I_k$2 for all $I_k$3, i.e., whether every interval $I_k$4 contains a prime. Stronger numerical evidence suggests $I_k$5 for all $I_k$6, and further theoretical work is invited to sharpen these lower bounds or adapt the exact formula to shorter intervals relevant for Legendre's conjecture proper.

The possibility of extending the matrix-theoretic method to intervals of the form $I_k$7 or $I_k$8 for $I_k$9 is explicitly raised, as is the challenge of finding asymptotic lower bounds for $B$0 surpassing $B$1 by a quantifiable margin.

Conclusion

This paper establishes an exact formula for counting primes in intervals between consecutive odd squares, founded on matrix-theoretic multiplicities and divisor analysis, circumventing direct primality testing. The work reframes classical questions about primes in short intervals within a combinatorial and divisor-theoretic context, exposes connections to limitations of traditional sieve theory, and points to a suite of open problems whose resolution may yield new insights into the distribution of primes in short intervals.