Pith Number

pith:K5YGYCGI

pith:2012:K5YGYCGIDJGPCXAWMFOBLDCQQ7

not attested not anchored not stored refs pending

A Refinement of Ramanujan's Factorial Approximation

Mark B. Villarino, Michael D. Hirschhorn

arxiv:1212.1428 v1 · 2012-11-27 · math.CA

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{K5YGYCGIDJGPCXAWMFOBLDCQQ7}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

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.

Receipt and verification

First computed	2026-05-18T03:39:03.976917Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4

Aliases

arxiv: 1212.1428 · arxiv_version: 1212.1428v1 · doi: 10.48550/arxiv.1212.1428 · pith_short_12: K5YGYCGIDJGP · pith_short_16: K5YGYCGIDJGPCXAW · pith_short_8: K5YGYCGI

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/K5YGYCGIDJGPCXAWMFOBLDCQQ7 \
  | 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: 57706c08c81a4cf15c16615c158c5087e94dcbb98f6d568d3da991c7f57123a4

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "d69aba5e0416736503a0046166c2870c5102fefe6871c44d404de32a05222ba9",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.CA",
    "submitted_at": "2012-11-27T17:07:49Z",
    "title_canon_sha256": "a2abd004ef8b7c6081100a399aa1c562eb7c84d4b95dd33765c35029763624bf"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1212.1428",
    "kind": "arxiv",
    "version": 1
  }
}