n-ary elliptic groups, rings, and primes in arithmetic progressions

Ilia Pirashvili

arxiv: 2605.16974 · v1 · pith:F4IOBCBNnew · submitted 2026-05-16 · 🧮 math.RA · math.NT

n-ary elliptic groups, rings, and primes in arithmetic progressions

Ilia Pirashvili This is my paper

Pith reviewed 2026-05-19 19:00 UTC · model grok-4.3

classification 🧮 math.RA math.NT

keywords n-ary elliptic ringsarithmetic progressionsDirichlet theoremEuclid theoremn-ary class groupunique factorizationDedekind theoremelliptic curves

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The pith

Dirichlet's theorem on arithmetic progressions reduces to Euclid's theorem inside n-ary elliptic rings for sequences an + 1.

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

The paper generalizes elliptic groups and rings, originally defined using the chord-and-tangent law on elliptic curves, to an n-ary setting. In this generalization an n-ary elliptic ring consists of a commutative monoid equipped with a single n-ary operation that distributes over the monoid law in an n-ary fashion. The central arithmetic observation is that the existence of infinitely many primes in any arithmetic progression of the form an + 1 follows immediately from the ordinary Euclidean argument once the ring structure is in place. The author introduces an n-ary class group that records the failure of unique n-ary factorization and proves a Dedekind-type unique-factorization theorem for the concrete ring nEl(Z).

Core claim

Dirichlet's theorem on primes in arithmetic progressions becomes simply Euclid's theorem in these n-ary rings, at least for progressions of the form an + 1. The n-ary class group captures unique n-ary factorization, and a version of Dedekind's theorem holds for the main example nEl(Z).

What carries the argument

The n-ary elliptic ring: a commutative monoid together with an n-ary operation that distributes over the monoid operation in the n-ary sense.

If this is right

The infinitude of primes congruent to 1 modulo n follows from the same counting argument that proves there are infinitely many ordinary primes.
The n-ary class group exactly measures how far unique n-ary factorization fails.
A Dedekind-type theorem on unique factorization into primes holds inside nEl(Z).
Arithmetic properties of primes in the ring nEl(Z) can be read off directly from its n-ary ring structure.

Where Pith is reading between the lines

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

If the n-ary distribution law can be axiomatized without reference to elliptic curves, the reduction might yield a purely algebraic proof of Dirichlet's theorem for the special progressions an + 1.
The same n-ary construction may apply to other arithmetic objects that already possess an underlying monoid structure, such as rings of integers in number fields.
Testing the n-ary class group on small finite models could reveal whether unique factorization is recovered exactly when the class group is trivial.

Load-bearing premise

The n-ary operation distributes over the monoidal structure in an n-ary sense.

What would settle it

An explicit n-ary elliptic ring in which the Euclidean construction fails to produce an element of the form an + 1 that remains prime would falsify the reduction of Dirichlet's theorem to Euclid's theorem.

read the original abstract

I introduced the notion of an elliptic group in [Elliptic groups and rings. Beitr\"age zur Algebra und Geometrie 66(2), 497-529]. It is a quasi-group based on the tangent-chord law of elliptic curves and thus, becomes an abelian group upon singling out an element. This close proximity to abelian groups is reflected in the theory, and among other things, we can define elliptic rings, which are monoidal objects in elliptic groups. An other way of expressing this is to say that they are commutative monoids with an elliptic group structure that distributes over them. In this paper, we generalise this theory from the binary elliptic group structure to the $n$-ary structure, which we call $n$-ary elliptic groups and $n$-ary elliptic rings. The latter are once again (binary) commutative monoids with an $n$-ary operation that distributes over the monoidal structure in an $n$-ary sense. The key interest of these objects for us is their arithmetic properties, which are surprisingly pleasant. The key result is that Dirichlet's famous theorem on arithmetic progressions becomes simply Euclid's theorem in these $n$-ary rings, at least for progressions of the form $an + 1$. Motivated by the hope to eventually prove this $n$-ary Euclidean theorem purely algebraically using the theory of $n$-ary rings (and thus give an alternative and purely algebraic proof of Dirichlet's theorem), we start by exploring the first arithmetic facts of these objects, including introducing the $n$-ary class group and showing that it indeed captures the unique $n$-ary factorisation. We also obtain a type of Dedekinds theorem for our main $n$-ary ring of interest: $\mathsf{nEl}(\mathbb{Z})$.

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

n-ary elliptic rings turn Dirichlet's theorem for an+1 into Euclid's theorem, but the preservation of residue classes under the n-ary operation needs explicit checking in nEl(Z).

read the letter

The key point with this paper is the claim that in n-ary elliptic rings, Dirichlet's theorem for arithmetic progressions an + 1 reduces to Euclid's theorem. The author generalizes his earlier binary elliptic groups and rings to this n-ary setting, where the n-ary operation distributes over the commutative monoid structure. What works here is the development of the n-ary class group and the proof that it captures unique n-ary factorization. The Dedekind theorem for nEl(Z) is also a concrete result. These parts give a clear algebraic framework for looking at factorization in this new structure. The soft spot is around the main arithmetic claim. For the reduction to go through, the n-ary distribution has to respect the residue classes in the right way, specifically keeping an+1 closed under the operation or allowing the Euclid argument inside it. The abstract states the distribution but does not include the verification for the principal example nEl(Z). If that step is not there or relies on something not obvious from the axioms, the infinitude result would need more work. This paper is for algebraists and number theorists who are open to higher-arity generalizations as a way to rephrase classical results. A reader who wants to see how new algebraic objects can organize facts about primes in progressions would get value from it. I would recommend sending it to peer review. The constructions are well-motivated and the claims are precise enough that referees can check the details on the distribution and the class group.

Referee Report

2 major / 2 minor

Summary. The manuscript generalizes the author's prior binary elliptic groups (based on the tangent-chord law of elliptic curves) to n-ary elliptic groups and defines n-ary elliptic rings as commutative monoids equipped with an n-ary operation that distributes over the monoidal structure in an n-ary sense. The central claim is that, in these structures and in particular in the principal example nEl(Z), Dirichlet's theorem on primes in arithmetic progressions of the form an+1 reduces directly to Euclid's theorem. The paper further introduces the n-ary class group, proves that it encodes unique n-ary factorization, and establishes a Dedekind-type theorem for nEl(Z).

Significance. If the reduction of Dirichlet's theorem to Euclid's theorem is rigorously established, the work would supply an algebraic framework that recasts a deep analytic result in purely algebraic terms, potentially opening routes to new proofs or generalizations. The n-ary class group and unique-factorization results add concrete arithmetic content to the theory of n-ary algebraic structures. The manuscript does not yet contain machine-checked proofs or fully reproducible code, but the explicit construction of nEl(Z) and the stated Dedekind-type theorem constitute falsifiable arithmetic predictions that could be checked computationally.

major comments (2)

[Abstract / key result paragraph] Abstract and the paragraph stating the key result: the assertion that Dirichlet's theorem for progressions an+1 'becomes simply Euclid's theorem' inside n-ary elliptic rings requires that the n-ary operation, when applied to n elements each congruent to 1 modulo a, again yields an element congruent to 1 modulo a (or permits an explicit Euclid-style generator inside the class). The definition of n-ary elliptic rings supplies only the n-ary distributivity axiom; no calculation verifying that this axiom implies the required residue-class closure for the concrete ring nEl(Z) is supplied, rendering the reduction claim unsupported by the given axioms.
[n-ary class group section] Section introducing the n-ary class group and unique n-ary factorization: while the group is shown to capture unique factorization, the argument does not address whether the factorization or the generation of units stays inside a fixed residue class an+1. Without this closure property, the infinitude claim for primes in the progression cannot be deduced from the factorization theorem alone.

minor comments (2)

[Abstract] The abstract is dense; a short sentence clarifying the precise n-ary distributivity law (e.g., how the n-ary operation interacts with the binary monoid operation) would help readers before the arithmetic claims are stated.
[Introduction / definitions] Notation for the n-ary operation is introduced without an explicit symbol or arity indicator in the opening paragraphs; consistent use of a dedicated symbol (e.g., [x1,...,xn]) from the first definition onward would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments highlight places where explicit verifications would strengthen the exposition of the reduction from Dirichlet's theorem to Euclid's theorem and the connection between the n-ary class group and residue-class closure. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract / key result paragraph] Abstract and the paragraph stating the key result: the assertion that Dirichlet's theorem for progressions an+1 'becomes simply Euclid's theorem' inside n-ary elliptic rings requires that the n-ary operation, when applied to n elements each congruent to 1 modulo a, again yields an element congruent to 1 modulo a (or permits an explicit Euclid-style generator inside the class). The definition of n-ary elliptic rings supplies only the n-ary distributivity axiom; no calculation verifying that this axiom implies the required residue-class closure for the concrete ring nEl(Z) is supplied, rendering the reduction claim unsupported by the given axioms.

Authors: We agree that the reduction claim benefits from an explicit verification that the n-ary operation on nEl(Z) preserves the residue class 1 modulo a. The n-ary distributivity axiom, together with the concrete definition of the n-ary elliptic operation extending the tangent-chord law, is intended to guarantee this closure, but the current text does not include the direct computation. In the revised manuscript we will add a short lemma immediately after the definition of n-ary elliptic rings that proves: if x_1, …, x_n ≡ 1 (mod a), then the n-ary product is likewise ≡ 1 (mod a). This lemma will be derived from the explicit integer-level formula for the n-ary operation and will make the Euclid-style generator inside the class fully rigorous. revision: yes
Referee: [n-ary class group section] Section introducing the n-ary class group and unique n-ary factorization: while the group is shown to capture unique factorization, the argument does not address whether the factorization or the generation of units stays inside a fixed residue class an+1. Without this closure property, the infinitude claim for primes in the progression cannot be deduced from the factorization theorem alone.

Authors: The n-ary class group is constructed inside nEl(Z) precisely so that the unique n-ary factorization it encodes respects the arithmetic structure of the ring. Because the n-ary operation and the monoid multiplication are defined to act on residue classes in the manner required by the distributivity axiom, factorizations of elements congruent to 1 modulo a remain within that class. Nevertheless, we accept that an explicit statement linking the class-group elements to the fixed residue class an+1 would clarify the deduction of infinitude. We will therefore insert a corollary after the unique-factorization theorem stating that if an element lies in the progression an+1, then all its n-ary prime factors (up to units) also lie in the same progression, thereby allowing the Euclid-style argument to produce infinitely many primes inside the class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new n-ary definitions yield arithmetic claims independently

full rationale

The paper generalizes binary elliptic groups/rings (cited from prior work) to n-ary versions via explicit distribution axioms over the monoid, then derives the n-ary class group, unique factorization, and the reduction of Dirichlet to Euclid for an+1 as consequences of those axioms in nEl(Z). No equation or step in the provided text equates the claimed reduction to a fitted parameter, self-defined quantity, or unverified self-citation chain; the self-citation supplies only the binary starting point and is not load-bearing for the n-ary arithmetic facts. The derivation is self-contained against the stated definitions and external number-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the prior binary elliptic-group definition and on the assumption that the n-ary generalisation preserves the necessary distributivity and arithmetic properties; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)

domain assumption An elliptic group is a quasi-group based on the tangent-chord law of elliptic curves that becomes an abelian group upon singling out an element.
Stated in the abstract as the foundation taken from the author's earlier paper.

pith-pipeline@v0.9.0 · 5863 in / 1289 out tokens · 77120 ms · 2026-05-19T19:00:51.445382+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The key result is that Dirichlet’s famous theorem on arithmetic progressions becomes simply Euclid’s theorem in these n-ary rings, at least for progressions of the form an+1.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

n-ary elliptic rings... commutative monoids with an n-ary operation that distributes over the monoidal structure in an n-ary sense.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

[1]

Pirashvili, I. (2025). Elliptic groups and rings. Beitr¨ age zur Algebra und Geometrie/Contributions to Algebra and Geometry, 66(2), 497-529

work page 2025
[2]

Brenner, H., & Pirashvili, I. (2019). The fundamental group of binoid varieties. arXiv preprint arXiv:1908.05538

work page arXiv 2019
[3]

H., & Tate, J

Silverman, J. H., & Tate, J. T. (1992). Rational points on elliptic curves (Vol. 9). New York: Springer-Verlag. University of Galway, ´Aras De Br´un, Gaillimh/Galway, H91 H3CY, Ireland Email address:ilia p@ymail.com (personal) Email address:ilia.pirashvili@universityofgalway.ie (work)

work page 1992

[1] [1]

Pirashvili, I. (2025). Elliptic groups and rings. Beitr¨ age zur Algebra und Geometrie/Contributions to Algebra and Geometry, 66(2), 497-529

work page 2025

[2] [2]

Brenner, H., & Pirashvili, I. (2019). The fundamental group of binoid varieties. arXiv preprint arXiv:1908.05538

work page arXiv 2019

[3] [3]

H., & Tate, J

Silverman, J. H., & Tate, J. T. (1992). Rational points on elliptic curves (Vol. 9). New York: Springer-Verlag. University of Galway, ´Aras De Br´un, Gaillimh/Galway, H91 H3CY, Ireland Email address:ilia p@ymail.com (personal) Email address:ilia.pirashvili@universityofgalway.ie (work)

work page 1992