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arxiv: 2604.02386 · v1 · submitted 2026-04-02 · 🧮 math.GM

Recognition: 2 theorem links

· Lean Theorem

Exact Formulas for Coprime Representations of Even Integers Avoiding a Prime

Andres M. Salazar
Authors on Pith no claims yet

Pith reviewed 2026-05-13 21:08 UTC · model grok-4.3

classification 🧮 math.GM
keywords coprime representationseven integersprime avoidanceclosed-form expressionsremainder operatorcongruence solutionsEuclidean algorithmarithmetic progressions
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The pith

Explicit closed-form expressions exist for the number of coprime positive integer pairs summing to a given even integer while avoiding multiples of a fixed prime.

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

The paper derives exact formulas for g(2n,p), the count of pairs (h,k) with h+k=2n, h≤k, and both coprime to 6p for prime p≥5. The expressions are built from the remainder operator δ_k(x), step functions, and the smallest solutions to the congruences 6x≡-1 mod p and 6x≡-5 mod p, found via the Euclidean algorithm. A reader would care because the formulas turn an O(n) enumeration task into O(1) arithmetic after a short precomputation for any fixed p. They further establish that g(2n,p) is piecewise linear in n, with breaks at arithmetic progressions controlled by residues modulo 3 and p.

Core claim

The central claim is that g(2n,p) admits explicit closed-form expressions built from the canonical remainder operator δ_k(x)=x-k⌊x/k⌋, elementary step functions, and the minimal positive solutions of the congruences 6x≡-1 (mod p) and 6x≡-5 (mod p). These solutions are obtained directly from the Euclidean algorithm applied to the auxiliary equation δ_k(a0 x)=b0, which identifies the residue classes excluded by the coprimality conditions. The resulting formulas are piecewise affine along arithmetic progressions of n determined by residue classes modulo 3 and p.

What carries the argument

The canonical remainder operator δ_k(x) combined with the minimal solutions to 6x≡-1 (mod p) and 6x≡-5 (mod p) obtained via the Euclidean algorithm on δ_k(a0 x)=b0, which directly mark the excluded residue classes for gcd(h,6p)=gcd(k,6p)=1.

If this is right

  • For any fixed prime p, g(2n,p) evaluates in constant time once two residue parameters are precomputed in O(log p) steps.
  • The count g(2n,p) changes linearly within each arithmetic progression of n whose common difference divides 3p.
  • Direct enumeration of qualifying pairs, which scales linearly with n, is replaced by arithmetic operations independent of n.
  • The formulas match exhaustive counts exactly for all 2n up to 10^5 and all tested primes from 5 to 23.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Euclidean-algorithm technique for minimal solutions could be applied to count coprime pairs avoiding products of several small primes.
  • The piecewise-affine structure may allow closed-form summation formulas for the total number of such representations across many even integers.
  • One could test whether analogous remainder-based expressions exist for the unrestricted Goldbach problem or for sums with additional congruence constraints.

Load-bearing premise

The minimal solution produced by the Euclidean algorithm for the remainder equation correctly identifies every residue class that must be excluded to enforce the two coprimality conditions.

What would settle it

Computing both the closed-form value and the brute-force enumeration of pairs for a prime p larger than 23 and some 2n larger than 10^5, then finding any discrepancy between the two results.

Figures

Figures reproduced from arXiv: 2604.02386 by Andres M. Salazar.

Figure 1
Figure 1. Figure 1: shows the graphs (2n, g(2n, p)) for p ∈ {5, 7, 11} and 2n ≤ 105 . The piecewise affine structure governed by residue classes modulo 3 and p is clearly visible; the parallel affine families correspond to the residue classes of n modulo 3p, consistent with Theorem 4.13 [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
read the original abstract

Fix a prime $p \ge 5$ and define $g(2n,p)=\#\{(h,k)\in\mathbb{Z}_{>0}^2 : h+k=2n,\; h\le k,\; \gcd(h,6p)=\gcd(k,6p)=1\}$. We derive explicit closed-form expressions for $g(2n,p)$ in terms of the canonical remainder operator $\delta_k(x)=x-k\lfloor x/k\rfloor$, elementary step functions, and the minimal solutions of the congruences $6x \equiv -1 \pmod{p}$ and $6x \equiv -5 \pmod{p}$. A key ingredient is an explicit formula for the minimal solution of $\delta_k(a_0 x)=b_0$ obtained via the Euclidean algorithm, which determines the excluded residue classes directly. The resulting formulas show that $g(2n,p)$ is piecewise affine along arithmetic progressions of $n$, governed by residue classes modulo $3$ and $p$. For fixed $p$, after precomputing two residue parameters in $O(\log p)$ time, each evaluation of $g(2n,p)$ requires only $O(1)$ operations, compared to $O(n)$ for direct enumeration. The formulas are validated computationally for all $2n \le 10^5$ and primes $p \in \{5,7,11,13,17,19,23\}$, with perfect agreement with brute-force enumeration.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript derives explicit closed-form expressions for g(2n,p), the number of pairs (h,k) of positive integers summing to 2n with h ≤ k and both coprime to 6p (p prime ≥5), expressed via the remainder operator δ_k(x), elementary step functions, and minimal positive solutions to the congruences 6x ≡ -1 (mod p) and 6x ≡ -5 (mod p) obtained from the Euclidean algorithm. The resulting formulas are piecewise affine along arithmetic progressions of n with periods dividing 3p, enabling O(1) evaluation after O(log p) precomputation of two residue parameters. Computational validation shows exact agreement with brute-force enumeration for all 2n ≤ 10^5 and p in {5,7,11,13,17,19,23}.

Significance. If the closed-form expressions are rigorously correct, the work supplies an efficient, exact method for counting such coprime representations, improving on O(n) enumeration and enabling asymptotic or distributional analysis in analytic number theory. The parameter-free, Euclidean-derived construction and computational reproducibility are strengths that would make the formulas useful for further study of Goldbach-type problems with coprimality constraints.

major comments (2)
  1. [derivation of minimal solution formula] The section presenting the explicit formula for the minimal solution of δ_k(a0 x)=b0 (derived via the Euclidean algorithm) does not include a separate verification that every solution to the coprimality conditions gcd(h,6p)=gcd(k,6p)=1 is captured exactly once when 6 divides the step-function arguments. If an arithmetic progression is missed, the subtraction of excluded residue classes would be incomplete and the piecewise-affine formula for g(2n,p) would fail on that progression.
  2. [closed-form expression for g(2n,p)] The central claim that g(2n,p) equals the stated combination of step functions and minimal solutions rests on the assumption that the two congruences locate all excluded classes; however, no independent proof or exhaustive case analysis is supplied showing that the remainder operator interacts correctly with the factor 6 in all residue classes modulo 6p.
minor comments (2)
  1. [abstract] The abstract and introduction should explicitly state the precise range of p (e.g., p ≥ 5) and the definition of the canonical remainder operator δ_k(x) at first use.
  2. [computational validation] Table or figure captions for the computational validation should list the exact primes tested and the maximum 2n value to allow direct replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and for recognizing the potential utility of the closed-form expressions. We address the two major comments point by point below, providing clarifications on the underlying derivations while committing to strengthen the manuscript with additional verification as requested.

read point-by-point responses

  1. Referee: [derivation of minimal solution formula] The section presenting the explicit formula for the minimal solution of δ_k(a0 x)=b0 (derived via the Euclidean algorithm) does not include a separate verification that every solution to the coprimality conditions gcd(h,6p)=gcd(k,6p)=1 is captured exactly once when 6 divides the step-function arguments. If an arithmetic progression is missed, the subtraction of excluded residue classes would be incomplete and the piecewise-affine formula for g(2n,p) would fail on that progression.

    Authors: We agree that an explicit verification lemma would improve rigor and clarity. The Euclidean algorithm applied to 6x ≡ -1 (mod p) and 6x ≡ -5 (mod p) yields the minimal positive solutions because gcd(6,p)=1 for p≥5, ensuring 6 is invertible modulo p and that the resulting arithmetic progressions exactly identify the excluded classes where h or k shares a prime factor with 6p. The step functions are then defined to subtract these classes precisely once per period dividing 3p. To address the concern directly, we will insert a new lemma in the revised manuscript proving that the construction captures every valid pair exactly once, by verifying that the combined period 3p enumerates all residue classes modulo 6p and that the coprimality conditions are fully accounted for without overlap or omission. revision: yes

  2. Referee: [closed-form expression for g(2n,p)] The central claim that g(2n,p) equals the stated combination of step functions and minimal solutions rests on the assumption that the two congruences locate all excluded classes; however, no independent proof or exhaustive case analysis is supplied showing that the remainder operator interacts correctly with the factor 6 in all residue classes modulo 6p.

    Authors: The remainder operator δ_k interacts with the factor 6 through the explicit solutions of the two linear congruences, which isolate the residues modulo p that correspond to violations of gcd(h,6p)=1 or gcd(k,6p)=1 while respecting h+k=2n. Because the formulas are built from these minimal solutions and the step functions enforce the inequalities h≤k, the interaction holds uniformly across residue classes modulo 6p. Nevertheless, we acknowledge that an independent exhaustive verification would be valuable. In the revision we will add a short appendix providing a direct case-by-case check for the smallest primes (p=5 and p=7) that confirms the formula reproduces the exact count in every residue class modulo 6p, thereby supplying the requested independent confirmation. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent Euclidean algorithm on standard congruences

full rationale

The paper constructs g(2n,p) from an explicit formula for minimal solutions of δ_k(a0 x)=b0 obtained directly via the Euclidean algorithm applied to the congruences 6x ≡ -1 (mod p) and 6x ≡ -5 (mod p). This is a standard, parameter-free procedure independent of the final count formula. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear; the piecewise-affine expressions follow from subtracting the identified arithmetic progressions using elementary step functions. Computational checks for small n and p serve only as validation, not as input to the derivation. The chain is self-contained against external modular arithmetic.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard number-theoretic tools (Euclidean algorithm for minimal solutions, properties of the remainder operator) with no free parameters or invented entities visible in the abstract.

axioms (1)
  • standard math The Euclidean algorithm yields the minimal positive solution to the linear congruence δ_k(a0 x) = b0
    Invoked to determine excluded residue classes for the coprimality conditions.

pith-pipeline@v0.9.0 · 5569 in / 1216 out tokens · 37648 ms · 2026-05-13T21:08:26.477640+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    Halberstam and H.-E

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    American Mathe- matical Society, 2004

    Henryk Iwaniec and Emmanuel Kowalski.Analytic Number Theory. American Mathe- matical Society, 2004

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    Nathanson.Additive Number Theory: The Classical Bases

    Melvyn B. Nathanson.Additive Number Theory: The Classical Bases. Springer, 2000. 19

    work page 2000