Pith Number

pith:F4IOBCBN

pith:2026:F4IOBCBNUPJPTY3DR6XTCEEO2Q

not attested not anchored not stored refs resolved

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

Ilia Pirashvili

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

arxiv:2605.16974 v1 · 2026-05-16 · math.RA · math.NT

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\usepackage{pith}
\pithnumber{F4IOBCBNUPJPTY3DR6XTCEEO2Q}

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Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

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.

C2weakest assumption

The n-ary operation distributes over the monoidal structure in an n-ary sense, allowing the arithmetic properties (including reduction of Dirichlet's theorem to Euclid's) to hold in the defined n-ary elliptic rings.

C3one line summary

Introduces n-ary elliptic groups and rings in which Dirichlet's theorem on arithmetic progressions reduces to Euclid's theorem for an+1 progressions, while defining an n-ary class group that captures unique factorization and proving a Dedekind-type theorem for nEl(Z).

References

3 extracted · 3 resolved · 0 Pith anchors

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

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

[3] 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:i 1992

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:03:33.818827Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

2f10e0882da3d2f9e3638faf31108ed4099fda8a285d15d0117da3c95472eddb

Aliases

arxiv: 2605.16974 · arxiv_version: 2605.16974v1 · doi: 10.48550/arxiv.2605.16974 · pith_short_12: F4IOBCBNUPJP · pith_short_16: F4IOBCBNUPJPTY3D · pith_short_8: F4IOBCBN

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/F4IOBCBNUPJPTY3DR6XTCEEO2Q \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 2f10e0882da3d2f9e3638faf31108ed4099fda8a285d15d0117da3c95472eddb

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "3c49512d71650bd79b397e8ac71c7e998128b5503fefd2bb29fa8c2874cf45f5",
    "cross_cats_sorted": [
      "math.NT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.RA",
    "submitted_at": "2026-05-16T12:51:51Z",
    "title_canon_sha256": "b9d95c8ed12b21f02baaa35ede73e46948cbbc2e5c5034a7db5bf222c62d828b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.16974",
    "kind": "arxiv",
    "version": 1
  }
}