Pith Number

pith:PLDU2V7U

pith:2026:PLDU2V7UH6L63O55XRGUPF25YV

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Real geometric transcendence for the Gamma function

Arshay Sheth

The x-axis is the only real algebraic curve in R² whose image under the Gamma function lies inside an algebraic curve.

arxiv:2605.15117 v1 · 2026-05-14 · math.NT

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Record completeness

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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

the x-axis is the only real algebraic curve in R² whose image via the Gamma function is contained in an algebraic curve

C2weakest assumption

the corresponding complex geometric transcendence result of Eterović, Padgett and Zhao (2025) holds and the base-change argument of Tamiozzo (2023) applies directly to this real setting

C3one line summary

The x-axis is the only real algebraic curve in R² whose Gamma image is contained in an algebraic curve.

References

12 extracted · 12 resolved · 0 Pith anchors

[1] Gregory Volfovich Chudnovsky, Algebraic independence of constants connected with the exponential and elliptic functions , Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698–701, 767 1976

[2] Henri Cohen, Number theory. Vol. II. Analytic and modern tools. https://link.springer.com/book/10.1007/978-0-387-49894-2, Graduate Texts in Mathematics, 240, Springer, New York, 2007 2007 · doi:10.1007/978-0-387-49894-2

[3] Lucia Di Vizio and Federico Pellarin, The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function https://arxiv.org/abs/2508.21237, preprint (2025), arXiv: 2508 2025

[4] Sebastian Eterovi\'c and Adele Padgett, Some equations involving the Gamma function https://pubs.ams.org/journals/tran/2025-378-05/S0002-9947-2025-09361-9, Trans. Amer. Math. Soc. 378 (2025), no. 5, 3 2025

[5] Sebastian Eterovi\'c, Adele Padgett, and Roy Zhao, Bialgebraic varieties of the Gamma function https://arxiv.org/pdf/2509.24982, preprint (2025), arXiv:2509.24982 2025

Receipt and verification

First computed	2026-05-17T21:40:25.717057Z
Last reissued	2026-05-17T21:57:19.053860Z
Builder	pith-number-builder-2026-05-17-v1
Signature	unsigned_v0
Schema	pith-number/v1.0

Canonical hash

7ac74d57f43f97edbbbdbc4d47975dc54e8bb6f8e3d082f42f7aaacfb60ee790

Aliases

arxiv: 2605.15117 · arxiv_version: 2605.15117v1 · pith_short_12: PLDU2V7UH6L6 · pith_short_16: PLDU2V7UH6L63O55 · pith_short_8: PLDU2V7U

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/PLDU2V7UH6L63O55XRGUPF25YV \
  | 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: 7ac74d57f43f97edbbbdbc4d47975dc54e8bb6f8e3d082f42f7aaacfb60ee790

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e80b206ebd04535a2267d8a30c020f23aeeaede13d9e1f8f64049592ea37625c",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-14T17:30:18Z",
    "title_canon_sha256": "e906b3ae397dd1af48d2c5fe58c59f580fd185307ccf4cd3b08384884cfe1227"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15117",
    "kind": "arxiv",
    "version": 1
  }
}