Pith Number

pith:VFJICXYQ

pith:2015:VFJICXYQW7SZZL7WRN4XENYNXZ

not attested not anchored not stored refs pending

Markoff Triples and Strong Approximation

Alex Gamburd, Jean Bourgain, Peter Sarnak

arxiv:1505.06411 v2 · 2015-05-24 · 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{VFJICXYQW7SZZL7WRN4XENYNXZ}

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	2026-05-18T01:27:54.058092Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

a952815f10b7e59caff68b7972370dbe59642270bee6d193b7ff38817da99049

Aliases

arxiv: 1505.06411 · arxiv_version: 1505.06411v2 · doi: 10.48550/arxiv.1505.06411 · pith_short_12: VFJICXYQW7SZ · pith_short_16: VFJICXYQW7SZZL7W · pith_short_8: VFJICXYQ

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/VFJICXYQW7SZZL7WRN4XENYNXZ \
  | 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: a952815f10b7e59caff68b7972370dbe59642270bee6d193b7ff38817da99049

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f13b895bb4059db38d55c43aaec6ed4e7fdc2576b373e6ce75edd089f8e8df3a",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2015-05-24T06:48:46Z",
    "title_canon_sha256": "0791e5cd6a83208b951a73968c32e276095ffaa6d42bd196d84e004c0fff0758"
  },
  "schema_version": "1.0",
  "source": {
    "id": "1505.06411",
    "kind": "arxiv",
    "version": 2
  }
}