Pith Number

pith:RYZDAD5Q

pith:2026:RYZDAD5QSIYC2K4UC64YESVRH5

not attested not anchored not stored refs resolved

On the shortest open cubic equations

Ashleigh Ratcliffe, Bogdan Grechuk

The equation 7x³ + 2y³ = 3z² + 1 has no integer solutions.

arxiv:2603.29831 v2 · 2026-03-31 · math.GM

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\usepackage{pith}
\pithnumber{RYZDAD5QSIYC2K4UC64YESVRH5}

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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

We use cubic reciprocity to prove that the equation 7x^3+2y^3=3z^2+1 has no integer solutions.

C2weakest assumption

Cubic reciprocity can be applied directly to this equation without additional unstated conditions or case distinctions that might allow solutions.

C3one line summary

Proves no integer solutions for 7x^3 + 2y^3 = 3z^2 + 1 and updates the list of shortest open cubic equations.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] Brauer groups of certain affine cubic surfaces.arXiv preprint arXiv:2509.16042, 2025 2025

[2] The decision problem for exponential Diophantine equations.Ann 1961

[3] A systematic approach to diophantine equations: open problems.arXiv preprint arXiv:2404.08518, 2024 2024

[5] Cubic diophantine equations: integer solutions beyond direct search.The American Mathematical Monthly, to appear, 2026 2026

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-20T00:00:37.378840Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

8e32300fb092302d2b9417b9824ab13f663e8c97202b1ad38bd2bd0fa239697d

Aliases

arxiv: 2603.29831 · arxiv_version: 2603.29831v2 · doi: 10.48550/arxiv.2603.29831 · pith_short_12: RYZDAD5QSIYC · pith_short_16: RYZDAD5QSIYC2K4U · pith_short_8: RYZDAD5Q

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/RYZDAD5QSIYC2K4UC64YESVRH5 \
  | 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: 8e32300fb092302d2b9417b9824ab13f663e8c97202b1ad38bd2bd0fa239697d

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7e7f78d3ccb6ce1f9d0c872760a6312fe26ee8336430ccc56cc3aff25e11656d",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GM",
    "submitted_at": "2026-03-31T14:52:42Z",
    "title_canon_sha256": "31f527698762332a438b952bf298e21b87c0aeaedecc1e30d1caf49de304d12f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.29831",
    "kind": "arxiv",
    "version": 2
  }
}