Pith Number

pith:ZCRBWROR

pith:2011:ZCRBWRORV57R656OGWWXEEWMTX

not attested not anchored not stored refs pending

Integral Apollonian circle packings and prime curvatures

Jean Bourgain

arxiv:1105.5127 v3 · 2011-05-25 · math.NT

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

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{ZCRBWRORV57R656OGWWXEEWMTX}

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	2026-05-18T04:04:44.788555Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

c8a21b45d1af7f1f77ce35ad7212cc9dd315042601324b25a41cf75c7469976e

Aliases

arxiv: 1105.5127 · arxiv_version: 1105.5127v3 · doi: 10.48550/arxiv.1105.5127 · pith_short_12: ZCRBWRORV57R · pith_short_16: ZCRBWRORV57R656O · pith_short_8: ZCRBWROR

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/ZCRBWRORV57R656OGWWXEEWMTX \
  | 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: c8a21b45d1af7f1f77ce35ad7212cc9dd315042601324b25a41cf75c7469976e

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "042e9b8ac03f62c3ed64bd2fb5801e6478272e263b528314003dd7157e12e405",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2011-05-25T19:49:59Z",
    "title_canon_sha256": "ff35967ed52da401ea0dd61a9a2319038d9422dd2ef2fcf4ffe50c4f4193918d"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1105.5127",
    "kind": "arxiv",
    "version": 3
  }
}