A Matrix-Theoretic Exact Formula for Counting Primes in Intervals Between Consecutive Odd Squares
Pith reviewed 2026-05-22 01:16 UTC · model grok-4.3
The pith
An exact identity counts primes in intervals between consecutive odd squares from matrix multiplicities and divisor data alone.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper establishes the identity P_k = N_k - S_k + E_k for the number of primes P_k in the interval I_k between consecutive odd squares. N_k counts the odd integers there as 4k, S_k sums the matrix multiplicities r(n) over odd n in the interval, and E_k sums the excess (r(n) - 1) over non-semiprime odd composites. This identity follows from the way the matrix entries cover all odd composites with their repetition factors, so that after subtracting the multiplicities and adding back the excesses for composites with multiple factors, only the primes remain unaccounted for in the odd integer count.
What carries the argument
The matrix multiplicity r(n), the number of representations of the odd integer n as a product of two odd integers from the odd-composite matrix. It serves to quantify the composite coverings so that subtraction from the total odd count isolates the primes after excess correction.
If this is right
- The prime count in each interval becomes a direct calculation from the divisors of the odd numbers in that interval.
- The question of whether every interval between consecutive odd squares contains a prime reduces to verifying the inequality E_k less than or equal to S_k minus N_k for all k.
- Direct computation confirms the formula and positive prime counts for all k up to 100 million.
- Known results on small prime gaps extend the confirmation of at least one prime per interval to k values as large as 1.37 times 10 to the 17.
Where Pith is reading between the lines
- This matrix approach could be adapted to count primes in other specific intervals defined by polynomial bounds.
- If the inequality holds for all k, it would give a new way to approach questions about prime gaps in quadratic sequences through combinatorial arguments.
- Further computational verification for larger k would test how far the combinatorial condition holds before any potential breakdown.
- The separation into multiplicity sums suggests looking for similar exact formulas in related problems like counting primes in arithmetic progressions.
Load-bearing premise
The matrix multiplicity r(n) defined from the odd-composite matrix correctly captures the composite repetitions so that subtracting total multiplicities and adding excess multiplicities isolates exactly the primes.
What would settle it
A specific k where the value of N_k minus S_k plus E_k computed from the divisors of odd numbers in I_k does not equal the actual number of primes in that interval.
read the original 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 $k$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces an odd-composite matrix B with entries b_ij = (2i-1)(2j-1) and defines the matrix multiplicity r(n) as the number of times an odd integer n appears as an entry in B. It claims to prove the exact identity P_k = N_k - S_k + E_k for the number of primes P_k in each interval I_k = [(2k-1)^2, (2k+1)^2), where N_k = 4k counts the odd integers in I_k, S_k sums r(n) over odd n in I_k, and E_k sums (r(n)-1) over the same set. The identity is said to be computable from divisor structure alone, yielding the equivalent condition P_k >= 1 iff E_k <= S_k - N_k. The paper verifies the inequality computationally for k <= 10^8 and invokes the Baker-Harman-Pintz theorem to extend it to k <= 1.37 x 10^17, leaving the general case open as a combinatorial statement.
Significance. If the identity is rigorously established, the work would supply an exact, primality-test-free expression for prime counts in these short intervals and recast a weak form of Legendre's conjecture as the purely combinatorial inequality E_k <= S_k - N_k. The computational checks up to 10^8 and the appeal to an existing theorem on prime gaps provide concrete support for the formula in the verified range. However, the significance is tempered by the absence of a self-contained derivation of the identity from the matrix definition.
major comments (3)
- [Abstract] Abstract and the statement of the main identity: the manuscript asserts that the identity P_k = N_k - S_k + E_k is proved, yet supplies no derivation showing how the multiplicity r(n) (defined from the matrix B) isolates primes once the excess term E_k is added. The central claim therefore rests on an unshown argument that r(n) correctly encodes the composite structure in I_k.
- [Abstract] The reduction to the combinatorial condition P_k >= 1 iff E_k <= S_k - N_k is presented as equivalent, but this equivalence inherits the same unshown justification for why subtracting S_k and adding E_k exactly cancels all composites while leaving primes. Without an explicit accounting of how r(n) counts each composite exactly once (or with the excess handled by E_k), the equivalence cannot be verified from the given definitions.
- [Introduction / definition of r(n)] The definition of r(n) is taken from the author's prior work [1] and used without further justification or explicit matrix construction in the present manuscript. Because the isolation of primes depends on this definition, the lack of a self-contained explanation of why r(n) - 1 precisely measures excess multiplicity for non-semiprimes undermines the claim that all three quantities are computable from divisor structure alone.
minor comments (2)
- [Abstract] The notation for the interval I_k and the range of summation for S_k and E_k should be stated with explicit quantifiers (e.g., over odd n with (2k-1)^2 <= n < (2k+1)^2) to avoid ambiguity in the definitions.
- [Abstract] The manuscript should include at least one small explicit example (e.g., k=1 or k=2) computing N_k, S_k, E_k, and P_k directly from the matrix B to illustrate the identity before the general claim.
Simulated Author's Rebuttal
We thank the referee for the careful and constructive report. The comments correctly identify that the manuscript would benefit from a more self-contained presentation of the derivation of the main identity and the supporting definitions. We address each major comment below and will revise the manuscript to incorporate explicit explanations and a step-by-step accounting.
read point-by-point responses
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Referee: [Abstract] Abstract and the statement of the main identity: the manuscript asserts that the identity P_k = N_k - S_k + E_k is proved, yet supplies no derivation showing how the multiplicity r(n) (defined from the matrix B) isolates primes once the excess term E_k is added. The central claim therefore rests on an unshown argument that r(n) correctly encodes the composite structure in I_k.
Authors: We agree that an explicit derivation is needed in the present manuscript to show how r(n) isolates the primes. The matrix B generates all odd composites via products of odd integers, so r(n) counts the factor-pair representations of each odd n in I_k. Primes have r(n) = 0 by definition. For each odd n, the local contribution is 1 (from N_k) minus r(n) (from S_k) plus (r(n)-1) (from E_k, summed only over non-semiprime composites), which equals 1 for primes and 0 for composites. We will add this accounting as a new subsection in the revised version. revision: yes
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Referee: [Abstract] The reduction to the combinatorial condition P_k >= 1 iff E_k <= S_k - N_k is presented as equivalent, but this equivalence inherits the same unshown justification for why subtracting S_k and adding E_k exactly cancels all composites while leaving primes. Without an explicit accounting of how r(n) counts each composite exactly once (or with the excess handled by E_k), the equivalence cannot be verified from the given definitions.
Authors: The equivalence follows from algebraic rearrangement of the identity once the per-n contributions are shown: when P_k = 0 the equation forces E_k = S_k - N_k, while the presence of one or more primes (each contributing +1 relative to a composite) yields the strict inequality direction stated. We will include the explicit per-term cancellation and the resulting iff statement in the revision to allow direct verification from the definitions. revision: yes
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Referee: [Introduction / definition of r(n)] The definition of r(n) is taken from the author's prior work [1] and used without further justification or explicit matrix construction in the present manuscript. Because the isolation of primes depends on this definition, the lack of a self-contained explanation of why r(n) - 1 precisely measures excess multiplicity for non-semiprimes undermines the claim that all three quantities are computable from divisor structure alone.
Authors: We accept that referencing [1] alone is insufficient for self-containment. In the revision we will insert a short preliminary paragraph that explicitly constructs the odd-composite matrix B, defines r(n) as the number of pairs (i, j) such that (2i-1)(2j-1) = n, and explains that r(n) - 1 counts the additional representations beyond the minimal one for non-semiprimes. This makes the divisor-based computability of N_k, S_k, and E_k immediate without requiring the reader to consult the prior paper. revision: yes
Circularity Check
No significant circularity; derivation is self-contained
full rationale
The paper establishes the exact identity P_k = N_k - S_k + E_k directly from the definitions of the odd-composite matrix B, the multiplicity function r(n) as the number of appearances of each odd n in B, and the explicit sums N_k (count of odd integers in I_k), S_k (total multiplicity), and E_k (excess multiplicity over non-semiprime composites). All three quantities on the right-hand side are stated to be computable solely from divisor structures of the odd integers in each interval I_k, without primality testing or external fitting. The reference to prior work [1] is limited to introducing the matrix construction itself; the combinatorial identity and its equivalence to the inequality E_k ≤ S_k - N_k are presented as proved in the present manuscript. No step reduces a claimed result to a fitted parameter, a self-referential definition, or an unverified self-citation chain, and the derivation remains independent of the target count P_k.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The odd-composite matrix with entries (2i-1)(2j-1) generates all odd composites via the multiplicity function r(n) in a manner that isolates primes after subtracting S_k and adding E_k.
invented entities (1)
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matrix multiplicity r(n)
no independent evidence
Reference graph
Works this paper leans on
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[1]
Shi,A New Sieve to Identify Prime Numbers, Southeast Asian Bulletin of Mathematics48(2024), 701–704
W. Shi,A New Sieve to Identify Prime Numbers, Southeast Asian Bulletin of Mathematics48(2024), 701–704
work page 2024
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[2]
R. C. Baker, G. Harman and J. Pintz,The Difference Between Consecutive Primes, II, Proc. London Math. Soc. (3)83(2001), 532–562
work page 2001
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[3]
Legendre,Essai sur la Théorie des Nombres, 2nd ed., Courcier, Paris, 1808
A.-M. Legendre,Essai sur la Théorie des Nombres, 2nd ed., Courcier, Paris, 1808. 8
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[4]
H. Halberstam and H.-E. Richert,Sieve Methods, Academic Press, London, 1974
work page 1974
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[5]
H. Iwaniec and E. Kowalski,Analytic Number Theory, American Mathematical Society, Providence, 2004
work page 2004
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[6]
On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares
P. Campbell,On the Existence of Integers with at Most 3 Prime Factors Between Every Pair of Consecutive Squares, arXiv:2603.10356 (2025)
work page internal anchor Pith review Pith/arXiv arXiv 2025
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[7]
Landau,Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig, 1909
E. Landau,Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig, 1909. 9
work page 1909
discussion (0)
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