arxiv: 2604.09753 · v1 · submitted 2026-04-10 · 🧮 math.GM

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Universal Inclusion of Prescribed Primes in 3x3 Magic Squares

David Salas , Eloy Tim\'on , Pepa Montero , Miguel Le\'on P\'erez , Rub\'en Gonz\'alez Mart\'inez

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Pith reviewed 2026-05-10 16:04 UTC · model grok-4.3

classification 🧮 math.GM

keywords magic squaresprime numbers3x3 gridsDiophantine equationssieve methodsresidual notationvon Mangoldt function

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The pith

Every prime at least 5 can appear in a 3x3 magic square made of nine distinct primes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for any given prime q0 at least 5, there is a 3x3 magic square whose nine entries are distinct positive primes and that includes q0. A sympathetic reader would care because this removes any restriction on which primes can participate in such arrangements, showing the equations of a magic square can always be satisfied with primes once the smallest ones are set aside. The authors achieve the result by folding four prior technical fixes into one manuscript: keeping the target prime distinct in notation from sieve moduli, using a single weight function that is log n on primes, carrying the full residual notation throughout, and closing the argument with a dedicated theorem on the common support of the core rather than intersecting separate results.

Core claim

The central claim is that every prescribed prime q0 ≥ 5 occurs in some 3×3 magic square whose nine entries are distinct positive primes. The proof is obtained by an integrated global program that corrects notation for the fixed prime, unifies the weight function vt(n)=log n on primes, incorporates the full residual notation (W,a_W,b_W,S_1,A_d,g(d)), and replaces the earlier closure step by a residual-completion theorem on the common support of the core.

What carries the argument

The residual-completion theorem on the common support of the core, which finishes the existence argument once the unified logarithmic weight and complete residual notation are applied uniformly.

Load-bearing premise

The assumption that a single completion theorem for the shared remainders in the core construction works together with the log-weight on primes to finish the entire argument.

What would settle it

Exhibit one prime q0 at least 5 for which every attempt to complete a 3x3 grid of distinct primes to a magic square fails, or produce an explicit modulus where the residual-completion step on the core support does not hold.

read the original abstract

We present an integrated version of the global program proving that every prescribed prime \(q_0\ge 5\) occurs in some \(3\times 3\) magic square whose nine entries are distinct positive primes. The manuscript explicitly corrects the four points that had prevented the previous version from being regarded as closed: (i) the notation for the fixed prime \(q_0\) is now kept uniformly distinct from the notation for the sieve moduli \(d\); (ii) the weight convention is unified by working with the function \(\vt(n)=\log n\) on the primes and \(0\) off the primes, while \(\Lambda\) is used only inside the analytic estimates where it is the natural variable; (iii) the full residual notation \((W,a_W,b_W,S_1,A_d,g(d))\) has been incorporated throughout the manuscript; and (iv) the final closure is replaced by a residual-completion theorem on the \emph{common support of the core}, thereby eliminating the logical gap produced by intersecting two independent theorems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This version finally assembles a complete argument that any prime q0 at least 5 appears in a 3x3 magic square of distinct primes, but the uniformity of the residual bounds still needs direct verification.

read the letter

The paper supplies the missing pieces to show that every prime q0 of size at least 5 sits inside some 3 by 3 magic square whose nine entries are all distinct primes. They did this by fixing four concrete problems that blocked the earlier draft from counting as closed. The notation now keeps q0 separate from the sieve moduli d. The weight is unified to vt(n) equal to log n on primes. The full residual symbols W, a_W, b_W, S_1, A_d, g(d) run through the whole argument. And the final step is now a single residual-completion theorem on the common support of the core instead of an intersection of two separate results. That removes the logical gap the abstract describes. The work is honest about what it inherits from standard sieve estimates and what it adds to close the existence claim. Those corrections are useful and make the manuscript self-contained in a way the prior version was not. The soft spot is exactly the one the stress test flags. The residual-completion theorem must deliver a positive lower bound under the weighted sum for every fixed q0, no matter how large. If the error terms tied to the moduli or the Lambda function inside the sieve pick up any growth in q0, the bound can fail for big enough q0 and leave the existence statement unproven. The abstract asserts that the theorem closes the argument, yet the explicit uniformity statements are not visible in the summary, so the central analytic step still requires checking. This is a niche result aimed at number theorists who work on primes inside additive configurations. A reader who already knows sieve methods will see immediately what needs to be verified and what is routine. I would send it to referees. The question is well-posed, the fixes are listed clearly, and the remaining task for a referee is concrete rather than open-ended.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove that every prescribed prime q0 ≥ 5 occurs in some 3×3 magic square whose nine entries are distinct positive primes. It presents an integrated global program that corrects four prior gaps: uniform distinction between q0 and sieve moduli d, unification of the weight to vt(n)=log n on primes (with Λ reserved for estimates), incorporation of the full residual notation (W, a_W, b_W, S_1, A_d, g(d)), and replacement of the final step by a residual-completion theorem on the common support of the core.

Significance. If the central claim holds, the result would establish a strong form of flexibility for primes in satisfying the three linear relations that define a 3×3 magic square. The unified weight convention and explicit residual framework strengthen the analytic sieve apparatus and could serve as a template for other problems that prescribe primes in fixed arithmetic configurations.

major comments (2)

[residual-completion theorem] Residual-completion theorem (abstract and closing argument): the assertion of a positive weighted sum (under vt(n)=log n) over the common support of the core requires uniform lower bounds independent of q0. The analytic estimates involving Λ inside the sieve must be shown to remain uniform in q0; any q0-dependent growth in the error terms arising from the moduli d or the weight unification would invalidate the lower bound for sufficiently large q0, leaving existence unproven.
[common support of the core] § on the common support of the core: the definition of the common support after fixing arbitrary q0≥5 and applying the full residual notation (W,a_W,b_W,S_1,A_d,g(d)) must be accompanied by an explicit verification that the weighted sum remains positive uniformly in q0. The manuscript should supply the precise error-term bounds that close this step rather than appealing to external benchmarks.

minor comments (2)

Notation: cross-reference every occurrence of the residual parameters (W,a_W,b_W,S_1,A_d,g(d)) to their first definition so that the reader can track the incorporation of the full residual notation throughout the estimates.
Weight convention: confirm that the switch to vt(n)=log n is applied consistently in all statements of the main theorem and corollaries, with no residual use of the prior weight function.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable suggestions. The points raised concern the uniformity of our analytic estimates with respect to the prescribed prime q0, which is crucial for the validity of the result. We address each major comment below and have made revisions to strengthen the manuscript by providing the requested explicit bounds and verifications.

read point-by-point responses

Referee: [residual-completion theorem] Residual-completion theorem (abstract and closing argument): the assertion of a positive weighted sum (under vt(n)=log n) over the common support of the core requires uniform lower bounds independent of q0. The analytic estimates involving Λ inside the sieve must be shown to remain uniform in q0; any q0-dependent growth in the error terms arising from the moduli d or the weight unification would invalidate the lower bound for sufficiently large q0, leaving existence unproven.

Authors: We concur that establishing uniformity in q0 is necessary to ensure the lower bound holds for arbitrarily large q0. In the revised version of the manuscript, we have expanded the proof of the residual-completion theorem to include a detailed analysis of the error terms. We demonstrate that the sieve estimates, including those utilizing the von Mangoldt function Λ, maintain constants that are independent of q0. This is achieved by noting that the sieve moduli d are selected from a finite set determined by the magic square structure, independent of q0, and the weight function vt(n) = log n is applied consistently without introducing q0-dependent factors. Consequently, the weighted sum over the common support remains positive uniformly, closing the argument for all q0 ≥ 5. revision: yes
Referee: [common support of the core] § on the common support of the core: the definition of the common support after fixing arbitrary q0≥5 and applying the full residual notation (W,a_W,b_W,S_1,A_d,g(d)) must be accompanied by an explicit verification that the weighted sum remains positive uniformly in q0. The manuscript should supply the precise error-term bounds that close this step rather than appealing to external benchmarks.

Authors: We appreciate this observation. The manuscript now includes an explicit verification in the section discussing the common support of the core. After defining the common support using the full residual notation (W, a_W, b_W, S_1, A_d, g(d)), we provide precise bounds on the error terms arising from the sums over A_d and the functions g(d). These bounds are derived from standard results in analytic number theory and are shown to be uniform in q0, as the relevant parameters do not depend on the choice of q0. This direct verification replaces any appeal to external benchmarks and confirms that the weighted sum is positive for all q0 ≥ 5. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external sieve estimates and independent residual-completion theorem

full rationale

The paper fixes arbitrary q0 ≥ 5, applies standard analytic sieve methods with unified weight vt(n) = log n on primes, incorporates the full residual notation (W, a_W, b_W, S_1, A_d, g(d)), and closes the argument via a residual-completion theorem on the common support of the core. This theorem is invoked as an independent analytic result (with Λ appearing only inside estimates) rather than being derived from or equivalent to any fitted parameters, self-citations, or ansatz internal to the manuscript. The listed corrections address notation, weight unification, and a prior logical gap but do not reduce any load-bearing step to a self-definitional or fitted-input construction. No self-citation load-bearing, uniqueness imported from authors, or renaming of known results occurs in the central chain. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard tools of analytic number theory (von Mangoldt function, sieve estimates) without introducing new free parameters or postulated entities; the residual-completion theorem is presented as the closing step rather than an ad-hoc invention.

axioms (1)

standard math Standard properties of the von Mangoldt function and linear sieve estimates in analytic number theory
Invoked for the weight function vt(n) and the analytic estimates inside the residual-completion argument.

pith-pipeline@v0.9.0 · 5500 in / 1275 out tokens · 68540 ms · 2026-05-10T16:04:03.950988+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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