Pith Number

pith:V3XZGKOD

pith:2026:V3XZGKOD3RW6734PCGPPXB6MQV

not attested not anchored not stored refs resolved

Khayyam's Cubics and the Hidden Conic

Amir Asghari

Khayyam's proportional arguments for each cubic always produce an algebraically available third conic that stays geometrically unused.

arxiv:2605.13876 v1 · 2026-05-08 · math.GM

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\usepackage{pith}
\pithnumber{V3XZGKOD3RW6734PCGPPXB6MQV}

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3 Author claim open · sign in to claim

4 Citations open

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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

In each case, the construction yields not merely the two conics Khayyam uses, but a third algebraically available conic relation that remains geometrically unused.

C2weakest assumption

That Khayyam's proportional arguments generate these local conic relations such that a third conic is always algebraically present yet unused, without imposing modern coordinate concepts on the historical method.

C3one line summary

Khayyam's thirteen cubic constructions each contain an algebraically available but geometrically unused third conic, showing his method as a self-contained geometric algebra within classical Greek geometry.

References

8 extracted · 8 resolved · 0 Pith anchors

[1] Apollonius of Perga, Treatise on Conic Sections, edited by Thomas Little Heath, Cambridge University Press, Cambridge, 1896

[2] C. B. Boyer, The History of the Calculus and Its Conceptual Development, Dover, New York, 1959. MR0124178 1959

[3] T. L. Heath, A History of Greek Mathematics. Vol. I: From Thales to Euclid, Clarendon Press, Oxford, 1921 1921

[4] D. A. Kent and D. J. Muraki, A geometric solution of a cubic by Omar Khayyam \ in which colored diagrams are used instead of letters for the greater ease of learners, Amer. Math. Monthly 123 (2016), n 2016 · doi:10.4169/amer.math.monthly.123.2.149

[5] R. Rashed and B. Vahabzadeh, Omar Khayyam, the Mathematician, Bibliotheca Persica Press, New York, 2000 2000

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:19.243121Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

aeef9329c3dc6defef8f119efb87cc857a452ca26820755b15e5d4580ad64c27

Aliases

arxiv: 2605.13876 · arxiv_version: 2605.13876v1 · doi: 10.48550/arxiv.2605.13876 · pith_short_12: V3XZGKOD3RW6 · pith_short_16: V3XZGKOD3RW6734P · pith_short_8: V3XZGKOD

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/V3XZGKOD3RW6734PCGPPXB6MQV \
  | 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: aeef9329c3dc6defef8f119efb87cc857a452ca26820755b15e5d4580ad64c27

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "39048a9d90c95c7a39cd39db44541e749ba0a3571e58db81eff87aa4d5ec6b1b",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GM",
    "submitted_at": "2026-05-08T23:33:34Z",
    "title_canon_sha256": "c2343b877be374d53afdafed447300099592ed61cf4fb0090e96e02367131fd7"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13876",
    "kind": "arxiv",
    "version": 1
  }
}